Certified Finite Schemes for Linked SPX/VIX Markets

Shaosai Huang

doi:10.2139/ssrn.7435365

Working paper · September 2026

Certified Finite Schemes for Linked SPX/VIX Markets

Shaosai Huang

∗ Working paper. Comments welcome.

Kspectra Research Inc., Toronto, Canada · [email protected]

Abstract

A finite model of the linked SPX and VIX markets is normally justified by convergence to a named continuous target, and inherits that target’s constants. This paper certifies such models on themselves. At every resolution — a mesh, finite volatility atoms, a matrix and an initial law — the martingale row, the log-contract identity, the forward-budget tower and the density and Bayes rows are exact, log-spot marginals are finite Gaussian mixtures, and the volatility index is the log-contract row’s own window average, so the tie between the two markets holds before any quote is declared. Across resolutions, five computable matrix certificates assemble a refinement modulus bounding the adapted Wasserstein distance between neighbours, so its summability is a convergence criterion read off the matrices in use, in a regime where the contraction degenerates and the defect must be truncated. Under it the family is realized on one probability space and converges almost surely uniformly to a continuous Itô market carrying the same rows at a computed distance, its physical layer passing with no condition on the market price of risk. Revealing the atom path leaves every unconditional price intact but leaves a computable rolling-index defect, so conditional comparison is essential. Controlled jumps, a quantitative inverse and clock-preserving mark corrections match a declared capped panel exactly at one resolution without disturbing the family’s limit; only the atom chain moves a spot smile’s level. Leverage and marked correlation mark the architecture’s edges.

Keywords. SPX and VIX; finite volatility chain; adapted Wasserstein distance; bicausal coupling; refinement consistency; computable certificates; Gaussian mixture; log-contract identity; forward variance budget; exact panel matching.

MSC 2020. 91G20, 91G60, 60J10, 49Q22, 60B10, 60G44. JEL. G13, C63, C65, G12.

Contents

1Introduction
2The scheme and its certificates
3Exact structure at every resolution
4The refinement coupling
5Refinement rates and the limit
6What revelation costs
7Exact panel matching
1. 7.1Fixings closer than a tenor
8Certificates and refinement, computed
9Boundary test I: leverage
10Boundary test II: marked correlation and joint panels
11Extension experiments
12Conclusion and scope
AThe square-root construction
BProofs for exactification, leverage and correlation
References
How to cite

1Introduction

A finite linked SPX/VIX model is usually justified by convergence to a named continuous target: a variance chain carrying conditionally drawn returns, or a recombinant tree, is shown to converge to Heston or to SABR, and the error constants are that target’s [1, 21, 22, 32, 35, 37]. Here the finite chains are the objects and neighbouring resolutions are compared with one another: exactness is levelwise, consistency is interlevel, and the continuous market is an output rather than a hypothesis.

Which distance that comparison needs is settled. Marginal laws are too coarse for conditional readouts, and the repairs proposed independently in several fields—extended weak convergence [2], adapted distributions [28], the nested distance [41], causal and adapted Wasserstein distances [5, 10]—define one topology [6], metrized by an adapted Wasserstein distance $\mathcal {AW}_\rho$ over bicausal couplings. In it the space of filtered processes is Polish and its martingales are closed [8, 9]: an $\mathcal {AW}$ -Cauchy family has a limit, and that limit is again a market. What such a theorem presupposes is a way of deciding whether a given family is Cauchy. This paper supplies one, and every quantity in it is a finite computation on the matrices in use.

The second theme is the tie between the two markets. A joint calibration must respect that the index is the implied volatility of the log contract over its window, and the entropic and transport constructions that match SPX and VIX quotes exactly impose this on the fitted law as a constraint [11, 20, 25, 26]. In a resolution it is an identity, in force before any panel is declared, and the exactification below is built on it rather than against it.

1.1Main results

A resolution $r=(h,\cA ,P,\mu _0)$ is a mesh, a finite set of volatility atoms with variance labels $\nu$ , a stochastic matrix and an initial law; its chain moves the atom by $P$ and draws a Gaussian return whose variance is the label in force. Three statements carry the paper.

The rows are identities. Write $B$ for the forward budget field, $B(t_k;t_j,t_l):=h\sum _{m=j}^{l-1}(P^{\,m-k}\nu )_{y_k}$ , a matrix power against the label vector. For every mesh, atom set and kernel (Theorem 3.1),

\E \bigl [e^{X_{k+1}-X_k}\mid \cF _k\bigr ]=1,\qquad \E \bigl [X_l-X_k\mid \cF _k\bigr ]=-\tfrac 12B(t_k;t_k,t_l),\qquad U_{t_k}=\tfrac 1\Delta B(t_k;t_k,t_k+\Delta ),

and the density and Bayes rows are exact finite sums of Gaussian expectations. The spot is a martingale, log-spot marginals are finite Gaussian mixtures, and the index is the log-contract row’s own window average. No certificate enters.

Five finite computations bound an adapted distance. A resolution is certified by a Wasserstein contraction on adjacent atoms, a Lyapunov tail, an initial moment, and an exponential bound on its moves; a neighbouring pair adds a defect between $P$ and the square of the finer kernel, truncated at a level $\ell _m$ (Definition 2.4). A scheme asks these to be uniform, with a contraction budget per unit time, projectively compatible initial laws, and a level paying for its own excursions (Definition 2.6). The refinement coupling then gives (Lemma 4.4)

\varepsilon _m=\iota _{\rho ,m} +\frac {d_{\ell _m}+\zeta _m+\chi _m}{1-c_{m+1}^2}, \qquad \bigl \|\,y^{(m)}_k-y^{(m+1)}_{2k}\,\bigr \|_{L^\rho }\le \varepsilon _m\ \ (k\le n), \qquad \mathcal {AW}_\rho \bigl (y^{(m)},y^{(m+1)}\bigr )\le \varepsilon _m,

the last at the dates the two grids share, because that coupling is bicausal (Lemma 4.2) and the path metric is averaged over them. Thus $\sum _m\varepsilon _m<\infty$ is a matrix-level criterion for $\mathcal {AW}$ -Cauchyness, evaluated on two finite objects with no third process entering. Co-immersion carries the bound from payoffs to conditional readouts (Proposition 4.5), giving explicit rates for budgets, the rolling index, Lipschitz index and spot prices, and physical predictions with no restriction on the market price of risk (Theorem 5.3, Corollary 5.4, Proposition 5.5).

The limit, and exactification that preserves it. Under maximal weighted summability the family converges on one probability space, almost surely uniformly, to a continuous Itô process $X^\infty =x_0-\tfrac 12A^\infty +\int y^\infty \,dW$ carrying the same rows, with the distance to it equal to the tail of the same computed series (Theorem 5.6); the physical layer passes to the limit with $\E L^\infty =1$ and no integrability condition on $\lambda$ . On top of that, a declared finite panel within certified reach is matched exactly at one resolution: controlled jumps and a quantitative inverse solve capped index calls, including fixings closer than a tenor (Theorems 7.4 and 7.8), and a clock-preserving mark correction then matches capped spot calls without moving any index row (Theorem 10.9, Corollary 10.11). The block certificates keep every move probability in $(0,\tfrac 38]$ , and when the quotes are the family’s own limiting values the correction is charged into $\varepsilon _m$ , so the corrected family keeps the original limit (Proposition 7.5).

Two tests mark the edges. Conditioning cannot be dispensed with: revealing the atom path leaves the law of $(y,X)$ and every unconditional price untouched while leaving a rolling-index defect that is computed from $(P,\nu )$ and converges upward to a nonzero limit (Section 6). And the architecture is not indefinitely extensible: state-dependent leverage keeps the structural rows but can lose finite-mixture closure and forces a growing proxy range, while a marked correlation keeps Gaussian mixtures and the common-budget rows with a clock that separates from the predictable bracket (Sections 9 and 10).

1.2Contribution

(i): A convergence criterion decided on the matrices. The modulus (14) is assembled from $N-1$ quantile comparisons, two matrix–vector products and one matrix square, and it bounds $\mathcal {AW}_\rho$ between neighbouring resolutions, so that $\sum _m\varepsilon _m<\infty$ decides Cauchyness in the metric under which filtered processes form a Polish space with closed martingales. It is built for the regime refinement creates, in which the contraction factors approach one and a fixed ergodicity rate is unavailable: the budget is imposed per unit time (11), the comparison runs between different lattices and between $P$ and the square of the finer kernel, and the defect is truncated with the excursion charged at a level that pays for itself. Section 8 computes every entry and shows what the truncation carries: the all-atom variant of the same certificate grows under refinement, and the uniform-metric one fails to decay with the atom spacing, where the truncated one decays.
(ii): The finite form of the SPX/VIX tie. The index is the log-contract row’s own window average at every resolution, so the tie is in force before a panel is declared. The level of a spot smile is then the clock: a clock-preserving mark correction reaches shape alone, and level is moved only by the atom chain (Theorem 10.9, Remark 10.10), with that chart’s singular values computed in Section 11.
(iii): Exactification that keeps the limit. A declared capped-call panel within certified reach is matched exactly at a resolution by a correction supported on declared blocks, whose move probabilities the block-mass and slope certificates place in $(0,\tfrac 38]$ , and whose displacement is charged into the refinement modulus; a family reproducing its own limiting panel at every resolution therefore retains its limit (Theorem 7.4, Proposition 7.5). A quantitative inverse with computed constants treats fixings closer than a tenor (Theorem 7.8).
(iv): A limit with a mode and a rate. Theorem 5.6 realizes the limit on one probability space, with almost surely uniform paths, as a continuous Itô process carrying the resolutions’ own rows, at a distance equal to the tail of the series already computed. Its index converges uniformly although the window moves with the valuation date, and its physical layer passes with $\E L^\infty =1$ , free of any condition on $\lambda$ .

The first of these is not particular to a volatility chain. Its three ingredients—a bicausal coupling, a defect truncated with its excursion charged, and a contraction budget stated per unit time—ask only that neighbouring kernels be comparable, so the same certification is available to other refining families that have to be justified without naming a limit.

1.3Relation to existing work

The mechanism behind $\varepsilon _m$ is the one familiar from numerical analysis, stability from a coupling and consistency from a defect between neighbouring discretizations, with the synchronous quantile coupling attaining $W_\rho$ at each step. Certificate (C1) is the Wasserstein contraction defining the coarse Ricci curvature of Ollivier [39], and with (C2) it is the pairing used by Hairer and Mattingly [27]; the closest quantitative statement is that of Rudolf and Schweizer [43], whose bound for two kernels on one state space carries the same geometric factor with an ergodicity rate fixed in the time index, a one-step difference measured by a Lyapunov-weighted supremum over the whole space, and a conclusion for the laws at step $n$ . Refinement moves all three: the kernels here act on different lattices over different meshes, the contraction factors tend to one so that it is the budget per unit time that is uniform, the defect is truncated with the excursion charged separately—which Section 8 shows is what lets the estimate close, the all-atom defect rising under refinement where the truncated one falls—and the bound travels on a coupling, so it reaches conditional readouts. Convergence of filtrations, which that construction makes unnecessary, is the subject of Brémaud and Yor [12] and Coquet et al. [19], and Backhoff-Veraguas et al. [7] bound $\mathcal {AW}$ between the laws of two diffusions.

Mixtures enter the smile literature differently: Brigo and Mercurio [14] construct a one-factor local volatility diffusion whose marginals are mixtures of lognormals sharing one forward, with weights fixed at the initial date, and drawing a volatility at that date instead reproduces those marginals while changing every conditional law [13]—the distinction Section 6 turns into a number. A volatility state carrying its own forward budget, as here, is what makes the log-contract row and the budget tower available at all; the finite-state architecture itself is that of Naik [36] and Buffington and Elliott [18], and its refinement analysis that of Lo and Skindilias [34], Li and Zhang [33] and Zhang and Li [44]. Matching a declared panel exactly on a finite object is the implied-tree programme of Rubinstein [42] and Derman and Kani [24], the relative-entropy reweighting of Avellaneda et al. [4], and the discrete calibrated martingale of Buehler and Ryskin [17]; what Section 7 adds is a correction local to declared blocks, stochastic by certificate, and charged so that a refining family survives it. The structural rows and their certification are those of Huang [29]; the two-law reading of the physical layer is that of Huang [30].

1.4Organisation

Sections 2–5 follow the main line: scheme, exact structure, coupling, rates and limit. Section 6 is the revelation test and Section 7 the exact panel, which distinguishes the family’s own limiting quotes from external ones; Section 8 computes the base certificates. Sections 9 and 10 are Boundary Tests I and II, leverage and marked correlation, the latter carrying spot rows and the joint panel; Section 11 reports their experiments and Section 12 concludes.

2The scheme and its certificates

Fix a horizon $H$ , a tenor $\Delta$ and an operating horizon $T_{\rm op}$ with $T_{\rm op}+\Delta \le H$ . Every resolution below is finite, and each of its one-level certificates is a finite computation. A refining scheme additionally requires uniform constants and a summable sequence of those computed quantities. This section separates the two requirements.

2.1Resolutions, chains, readouts

Definition 2.1 (Resolution) . A resolution is a quadruple $r=(h,\cA ,P,\mu _0)$ in which

(i): $h>0$ is a mesh with $H/h\in \N$ and $\Delta /h\in \N$ , so that the tenor and the horizon are grid-aligned; write $n:=H/h$ , $t_k:=kh$ and $m_\Delta :=\Delta /h$ ;
(ii): $\cA =\{a_1<\dots <a_N\}\subset (0,\infty )$ is a finite set of volatility atoms, with variance labels $\nu _i:=a_i^2$ and spacing
$\begin{equation} \delta _r:=\max \bigl \{a_1,\ \max _{i<N}(a_{i+1}-a_i)\bigr \}; \label {eq:spacing} \end{equation}$
(iii): $P$ is a stochastic matrix on $\cA$ and $\mu _0$ a law on $\cA$ .

Definition 2.2 (The chain of a resolution) . Let $(y_k)_{k\le n}$ be the Markov chain with kernel $P$ and $y_0\sim \mu _0$ , and let $(\xi _k)_{k<n}$ be independent standard normals, independent of $(y_k)$ . The chain of $r$ is the pair $(y,X)$ with

\begin{equation} X_{k+1}-X_k:=-\tfrac 12h\,\nu _{y_k}+\sqrt {h\,\nu _{y_k}}\;\xi _k, \qquad X_0=x_0, \label {eq:return} \end{equation}

its spot is

S:=e^X

, and within a cell we interpolate

X_t:=X_k-\tfrac 12\nu _{y_k}(t-t_k)+\sqrt {\nu _{y_k}}\,(W_t-W_{t_k})

along a Brownian motion

W

carrying the increments

\sqrt h\,\xi _k

. Its filtration is

\cF _k:=\sigma (y_0,\dots ,y_k,\xi _0,\dots ,\xi _{k-1})

Conditionally on the atom path the returns are independent Gaussians, so every marginal of $X$ is a finite Gaussian mixture. That mixture structure closes the return-pricing layer; separately, the finite-state volatility chain makes budgets and volatility-index readouts finite matrix computations.

Definition 2.3 (Readouts) . With $\nu =(\nu _i)$ read as a column vector, the chain’s clock, budget field, rolling index and index are

\begin{equation} A_{t_k}:=h\sum _{i<k}\nu _{y_i},\qquad B(t_k;t_j,t_l):=h\sum _{m=j}^{l-1}\bigl (P^{\,m-k}\nu \bigr )_{y_k}\ \ (k\le j<l), \label {eq:budget} \end{equation}

\begin{equation} U_{t_k}:=\frac 1\Delta \,B(t_k;t_k,t_k+\Delta ) =\frac h\Delta \sum _{m<m_\Delta }\bigl (P^{\,m}\nu \bigr )_{y_k}, \qquad V_{t_k}:=\sqrt {U_{t_k}} . \label {eq:index} \end{equation}

For a market price of risk

\lambda \in \R

the density and the physical law are

\begin{equation} L:=\prod _{k<n}\exp \Bigl (\lambda \sqrt {h\nu _{y_k}}\,\xi _k-\tfrac 12\lambda ^2h\nu _{y_k}\Bigr ), \qquad \Pp :=L\,\Q , \label {eq:density} \end{equation}

\Q

being the law of Definition 2.2. Prices of bounded measurable payoffs of

(X,V)

at grid dates are conditional expectations under

\Q

, and physical predictions are conditional expectations under

\Pp

Each readout is a finite computation from $(P,\nu )$ : the budget field is a matrix power against $\nu$ , the index is a partial sum of such powers, a call on $V$ is a finite sum over atoms, a call on $S$ at a grid date is a finite mixture of Gaussian integrals in closed form, and $L$ is a product of Gaussian factors, so the Bayes rows are identities between finite sums of Gaussian expectations. No approximation is involved anywhere in this list, and nothing in it refers to a continuous model.

2.2The five certificates

The results of this paper rest on five properties of a resolution, or of a pair of resolutions. Each is a finite computation, which is why we call them certificates rather than assumptions: for a chain that has actually been built, they are checked, not postulated.

Write $F_{P(a,\cdot )}$ for the distribution function of the row $P(a,\cdot )$ and $F^{-1}_{P(a,\cdot )}$ for its quantile function, a step function with at most $N$ jumps.

Definition 2.4 (Certificates of a resolution) . Fix an order $\rho \ge 1$ . For laws on the line with quantile functions $F^{-1},G^{-1}$ write $W_\rho (\mu ,\nu )=\bigl (\int _0^1|F^{-1}(u)-G^{-1}(u)|^\rho du\bigr )^{1/\rho }$ , the Wasserstein distance of order $\rho$ , which the quantile coupling attains. Let $r=(h,\cA ,P,\mu _0)$ and let $c\in (0,1)$ , $\beta >0$ , $c_\beta \ge 0$ , $\vartheta >0$ .

(C1): Contraction at order $\rho$ with factor $c$ : the rows are stochastically ordered, and for adjacent atoms
$\begin{equation} W_\rho \bigl (P(a_{i+1},\cdot ),P(a_i,\cdot )\bigr )\ \le \ c\,(a_{i+1}-a_i), \qquad i<N . \label {eq:C1} \end{equation}$
(C2): Lyapunov tail with rate $\beta$ : with $V_\beta (a):=e^{\beta a^2}$ ,
$\begin{equation} (PV_\beta )(a)\ \le \ e^{c_\beta h}\,V_\beta (a)\qquad \text {for every atom }a . \label {eq:C2} \end{equation}$
(C3): Initial control: $\E _{\mu _0}V_\beta <\infty$ , which for a finite $\cA$ is automatic and quantitative.
(C4): Diffusive moves with rate $\vartheta$ : for every atom $a$ ,
$\begin{equation} \E _{P(a,\cdot )}\exp \Bigl (\frac {\vartheta \,(y_1-a)^2}{h}\Bigr )\le 2 . \label {eq:C4} \end{equation}$

For two resolutions $r=(h,\cA ,P,\mu _0)$ and $r'=(h/2,\cA ',P',\mu _0')$ , let $\pi :\cA '\to \cA$ and $\pi ':\cA \to \cA '$ be the nearest-atom maps. When $\mu _0=\pi _{\#}\mu '_0$ , define the projection errors actually used by the coupling,

\begin{equation} \iota _\rho (r,r'):=\Bigl (\E _{\mu '_0}|y-\pi (y)|^\rho \Bigr )^{1/\rho }, \qquad \zeta (r,r'):=\max _{a\in \cA }|a-\pi '(a)|, \label {eq:projection-errors} \end{equation}

and, for a level

\ell >0

, define the truncated refinement defect

\begin{equation} d_\ell (r,r'):=\max _{a\in \cA ,\ a\le \ell }\ W_\rho \bigl (P(a,\cdot ),\,(P')^2(\pi '(a),\cdot )\bigr ) . \label {eq:defect} \end{equation}

(C5): Consistency at level $\ell$ : $d_\ell (r,r')$ is small compared with the contraction budget, quantified through the refinement modulus (14) below.

Remark 2.5 (Why the order matters, and why adjacent atoms suffice) . Two features of (C1) are deliberate. First, it is checked on adjacent atoms only, and that is enough: $W_\rho$ obeys the triangle inequality, so summing (6) along the grid gives $W_\rho (P(x',\cdot ),P(x,\cdot ))\le c\,(x'-x)$ for every pair $x<x'$ , and applying it twice gives the factor $c^2$ for the two-step kernel $(P')^2$ . Second, the distance is an average over levels and not a supremum. In the uniform metric $W_\infty$ neither feature survives: adjacent slacks add rather than compose, so the factor over a general pair degrades with the number of atoms between them; and the supremum is attained at extreme levels, where two resolutions’ staircases can sit far apart on a set of levels of negligible measure, so it does not scale with the atom spacing at all. Section 8 reports both, and the difference is the difference between a certificate that decays and one that does not.

(C4) says the kernel moves the state by $O(\sqrt h)$ with Gaussian tails; it is what makes the limit of Section 5 continuous, and with a union bound it produces the factor $\sqrt h\,(1+\sqrt {\log 2n})$ in which every rate below is expressed. Each certificate is a finite computation: (C1) is $N-1$ comparisons of quantile functions, (C2) and (C4) are $N$ inequalities on a matrix–vector product, and (C5) is one matrix square and $N$ comparisons. None of them refers to a limit, a model, or an analytic family.

Definition 2.6 (Scheme) . A scheme of order $\rho$ is a sequence $\mathsf r=(r_m)_{m\ge 0}$ of resolutions with $h_{m+1}=h_m/2$ and the following five properties.

(S1): Nested ranges. $\min \cA _{m+1}\le \min \cA _m$ and $\max \cA _m\le \max \cA _{m+1}$ . The coarse-to-fine error $\zeta _m:=\zeta (r_m,r_{m+1})$ is computed from the two atom sets and is zero when $\cA _m\subset \cA _{m+1}$ . No spacing surrogate is used for a projection onto a cap.
(S2): Compatible initial laws. $\mu _{0,m}=(\pi _m)_{\#}\mu _{0,m+1}$ for every $m$ , together with $\sup _m\E _{\mu _{0,m}}V_\beta <\infty$ , the uniform form of (C3) that the constants below use. Write $\iota _{\rho ,m}:=\iota _\rho (r_m,r_{m+1})$ for the initial coupling error. Separate nearest-grid projections of one external law need not be compatible. Nested atom sets with one common initial atom, as in Section 8, give $\iota _{\rho ,m}=\zeta _m=0$ .
(S3): Uniform certificates. (C1)–(C4) hold at order $\rho$ with $\beta$ , $c_\beta$ and $\vartheta$ independent of $m$ , and with a common contraction budget: for some $\kappa _*>0$ ,
$\begin{equation} 1-c_m^2\ \ge \ \kappa _*\,h_m\qquad \text {for every }m . \label {eq:cbudget} \end{equation}$
The factors themselves tend to one — a finer mesh contracts less per step — so it is the budget per unit time, not the factor, that is uniform.
(S4): Levels that pay for their own excursions. A sequence of levels $\ell _m$ whose excursion charge
$\begin{equation} \chi _m:=\Xi _m\bigl (C_0\,e^{-\beta \ell _m^2}\bigr )^{1/\rho }, \qquad \Xi _m:=\max \cA _{m+1}, \label {eq:charge} \end{equation}$
with $C_0$ the constant of Lemma 5.1, obeys $\chi _m\le \tfrac 12\delta _{m+1}$ ; equivalently
$\begin{equation} \beta \,\ell _m^2\ \ge \ \log C_0+\rho \log \bigl (2\,\Xi _m/\delta _{m+1}\bigr ). \label {eq:level-rule} \end{equation}$
(S5): Summable modulus. The refinement modulus
$\begin{equation} \varepsilon _m:=\iota _{\rho ,m} +\frac {d_{\ell _m}(r_m,r_{m+1})+\zeta _m+\chi _m}{1-c_{m+1}^2} \qquad \text {satisfies}\qquad \sum _{m\ge 0}\varepsilon _m<\infty . \label {eq:refmod} \end{equation}$

The scheme is of rate $\gamma >0$ if $\varepsilon _m=O(h_m^{\gamma })$ .

The finer contraction gives the denominator $1-c_{m+1}^2\ge \kappa _*h_{m+1}$ . Its numerator is the one-step defect, the actual coarse-to-fine projection error, and the excursion charge. The last is incurred at every step and therefore belongs inside the geometric sum. Rule (13) makes it comparable with a grid error and gives the exact lower bound

\begin{equation} \ell _m\ \ge \ \left \{\frac 1\beta \left [\log C_0+ \rho \log \!\left (\frac {2\Xi _m}{\delta _{m+1}}\right )\right ]\right \}^{1/2}. \label {eq:level-growth} \end{equation}

This is not an upper bound: a scheme may choose a larger level or a faster growing cap. If $\log \Xi _m=O(\log (1/\delta _{m+1}))$ and $\ell _m$ is chosen within a fixed factor of the right-hand side, then $\ell _m\asymp \sqrt {(\rho /\beta )\log (1/\delta _{m+1})}$ ; under $\delta _m\asymp h_m^{3/2}$ this is $\ell _m\asymp \sqrt {(3\rho /(2\beta ))\log (1/h_m)}$ . The square-root construction below satisfies the cap condition and uses this near-minimal scale. In particular, its atom cap must grow with the level. The initial error $\iota _{\rho ,m}$ is outside the geometric sum because it is never amplified. Uniformity in (S3) and summability in (S5) are family assumptions; their entries are finite computations.

2.3An analytic construction

Lemma 2.7 (A square-root scheme of rate $\tfrac 12$ , at any fixed order) . Let $d_B:=4\kappa \theta /\sigma ^2$ be an integer with $d_B\ge 2$ , and let $K_t$ be the transition kernel of $dv=\kappa (\theta -v)dt+\sigma \sqrt v\,dZ$ in the coordinate $y=\sqrt v$ . Fix $\rho \ge 1$ , $\beta <2\kappa /\sigma ^2$ and $\eta >0$ . For $h_m=h_02^{-m}$ take

\begin{equation} \delta _m=\frac {\delta _0}{3^m}\le h_m^{3/2},\qquad \frac {\delta _m}{h_m^{3/2}}\longrightarrow 0, \qquad \beta \ell _m^2\ge \log C_0+\rho \log \frac {2\Xi _m}{\delta _{m+1}}, \label {eq:sqrt-budgets-a} \end{equation}

and increasing caps with

Y_m=O(\sqrt {\rho \log (1/h_m)})

and

\Xi _m:=Y_{m+1}

, satisfying

\begin{equation} Y_m-\ell _m\ \ge \ \sigma \sqrt {\frac {(1+\eta )\rho h_m}{2} \log \frac {4\Xi _m}{\delta _{m+1}}}. \label {eq:sqrt-budgets} \end{equation}

Let

\cA _m=\{\delta _m,2\delta _m,\ldots ,Y_m\}

, with each cap a lattice point. Write

q_m

for nearest-atom quantization with the two exterior cells sent to the end atoms, set

P_m(a,\cdot ):=(q_m)_{\#}K_{h_m}(a,\cdot )

, and choose one

\bar a\in \cA _0

with

\mu _{0,m}=\delta _{\bar a}

at every level. For

h_0

small enough,

r_m=(h_m,\cA _m,P_m,\mu _{0,m})

is a scheme of order

\rho

and rate

\tfrac 12

. More precisely,

(a): (C1) holds with $c_m\le e^{-\kappa h_m/(2\rho )}(1+o(h_m))^{1/\rho }$ and hence $1-c_m^2\ge \kappa h_m/(2\rho )$ ;
(b): (C2) holds with one common $c_\beta$ , and (C4) holds, for example, with the explicit common rate
$\begin{equation} \vartheta _*:=\frac {1}{4096\,\sigma ^2(1+d_B)}; \label {eq:sqrt-theta} \end{equation}$
(c): the truncated defect obeys
$\begin{equation} d_{\ell _m}(r_m,r_{m+1})\ \le \ \delta _m+3\delta _{m+1}; \label {eq:sqrt-defect} \end{equation}$
(d): $\iota _{\rho ,m}=\zeta _m=0$ and $\varepsilon _m=O(\rho \delta _m/h_m)=O(h_m^{1/2})$ .

The order enters through the level and cap. Truncation is essential: at a cap the two resolutions dispose of overflow on different atoms, and the all-atom defect need not decrease.

Remark 2.8 (One family serves every lower order) . $W_{\rho '}\le W_\rho$ for $\rho '\le \rho$ , so (C1) and the defect at order $\rho$ imply them at every lower order with the same constants, and the charge (12) only decreases; a scheme of order $\rho$ is therefore a scheme of every order $\rho '\le \rho$ . This is what makes the readouts of Theorem 5.3 available: that theorem reads the modulus at order $4\rho$ , so an $L^2$ statement about a readout asks for a scheme of order $8$ , and Lemma 2.7 supplies one — with caps about twice those the order- $2$ budgets would require.

A proof from the integer-dimensional Ornstein–Uhlenbeck representation is given in Appendix A.

The example is the whole role of the square-root process in the theory: it shows the certificates are satisfiable, with closed-form constants, and gives Section 8 a construction to compute them on. No general theorem assumes it.

Remark 2.9 (Certificates of a fitted chain) . The example generates its chains from a kernel, but nothing in Definition 2.6 requires that. A chain produced by calibration is a matrix, and its certificates are computed from that matrix: (C1) by comparing adjacent rows’ quantile functions, (C2) and (C4) by one matrix–vector product each, (C5) by squaring the finer matrix. The contraction rate $\kappa$ and the tail rate $\beta$ that appear in every constant below are therefore outputs of the chain in use, not properties assumed of a market.

3Exact structure at every resolution

The rows below are identities of the construction of Section 2. They hold at every mesh, every atom set and every kernel, with no certificate and no limit; only the quantitative statements of Sections 4–5 use (C1)–(C5).

Theorem 3.1 (Exact structure at every resolution) . Let $r=(h,\cA ,P,\mu _0)$ be a resolution and let $(y,X)$ be its chain. Under $\Q$ :

(a): $S=e^X$ is a strictly positive $(\cF _k)$ -martingale. Every marginal of $X$ is a finite Gaussian mixture, equivalently every marginal of $S$ is a finite lognormal mixture: conditionally on the atom path, $X_k-x_0\sim N(-\tfrac 12A_{t_k},A_{t_k})$ .
(b): The diagonal log-contract identity holds exactly: $\E [X_l-X_k\mid \cF _k]=-\tfrac 12B(t_k;t_k,t_l)$ for $k\le l$ .
(c): The budget field is nonnegative, additive in the window, $B(t_k;t_i,t_l)=B(t_k;t_i,t_r)+B(t_k;t_r,t_l)$ , and a martingale in the valuation index: $\E [B(t_{k'};t_i,t_l)\mid \cF _k]=B(t_k;t_i,t_l)$ for $k\le k'\le i$ .
(d): $\E L=1$ ; the density process is explicit, $D_k=\prod _{i<k}\exp (\lambda \sqrt {h\nu _{y_i}}\,\xi _i-\tfrac 12\lambda ^2h\nu _{y_i})$ ; under $\Pp$ the atom chain has the same law as under $\Q$ , and $X_{k+1}-X_k\sim N((\lambda -\tfrac 12)h\nu _{y_k},\,h\nu _{y_k})$ conditionally on $\cF _k$ . Consequently every Bayes row $\E _{\Pp }[G\mid \cF _k]=\E [LG\mid \cF _k]/D_k$ is an identity between finite sums of Gaussian expectations.

Proof.(a) Given $\cF _k$ the increment is $N(-\tfrac 12h\nu _{y_k},h\nu _{y_k})$ , whose exponential has conditional mean one; the mixture statement is the sum of independent Gaussian increments conditionally on the atom path. (b) Take conditional expectations in (2): the Gaussian terms vanish and the drift is $-\tfrac 12\E [A_{t_l}-A_{t_k}\mid \cF _k] =-\tfrac 12B(t_k;t_k,t_l)$ by (3) and the Markov property. (c) All three are properties of the conditional expectation of the nonnegative variable $A_{t_l}-A_{t_i}$ , additive in $(i,l)$ . (d) Conditionally on the atom path the exponent of $L$ is Gaussian with mean $-\tfrac 12\lambda ^2A_{t_n}$ and variance $\lambda ^2A_{t_n}$ , so $\E [L\mid y]=1$ ; the factors with index $i<k$ give $D_k$ , while the future factors integrate to one; integrating against the atom path leaves its law unchanged, and completing the square shifts the conditional mean of the return.∎

Remark 3.2 (What is finite, and what is exact) . The variance coordinate has $N$ states while the traded log-spot stays continuous-valued, so “finite” refers to the atom chain, to the log-spot’s finite Gaussian mixture and to the spot’s finite lognormal mixture at each grid date. The log-Euler architecture over a finite variance chain is standard [21, 22, 37]; what is new here is the certified comparison between resolutions and, under the conditions of Section 7, the controlled-jump exactification of a finite panel.

Remark 3.3 (Readouts are computations) . The atom chain makes budgets and index prices finite matrix computations; conditional Gaussian returns make grid-date spot prices finite mixtures of Gaussian integrals. Physical versions reweight the same finite path sums by $L$ . Thus the readouts are defined and computed at the resolution in use, without reference to a continuous target.

4The refinement coupling

Two resolutions of a scheme are compared on one probability space, by a coupling that reads the finer chain’s move and follows it with the coarser one. The comparison is between the two objects a user would actually compute, not between either of them and a model.

4.1The coupled space

Fix $m$ and write $r=(h,\cA ,P,\mu _0)$ for $r_m$ and $r'=(h/2,\cA ',P',\mu _0')$ for $r_{m+1}$ , with fine grid dates $t'_j=jh/2$ , so that $t'_{2k}=t_k$ . Let $W$ be a Brownian motion, let $(U'_j)$ be independent uniforms driving the fine atom chain, let $(\mathsf V_k)$ be further independent uniforms, and let all three families be independent.

Lemma 4.2 (It is a coupling, and both chains are co-immersed) . Under Definition 4.1, $(y,X)$ is the chain of $r$ and $(y',X')$ is the chain of $r'$ , each with its own law. Moreover $\cF _k\subseteq \cG _k$ and $\cF '_{2k}\subseteq \cG _k$ , and for every integrable $G$ measurable for the coarse chain’s whole path, $\E [G\mid \cG _k]=\E [G\mid \cF _k]$ , and likewise for the fine chain at fine dates.

Proof.Conditionally on $\cG _k$ the level $G_k$ of (20) is uniform on $(0,1)$ , because randomizing within the mass of the realized atom converts a discrete distribution function into a uniform variable, and it is independent of $\cG _k$ because $\mathsf V_k$ is. Hence $F^{-1}_{P(y_k,\cdot )}(G_k)$ has law $P(y_k,\cdot )$ given $\cG _k$ , which is both the Markov property of $y$ and its co-immersion in $(\cG _k)$ ; the returns are built from increments of $W$ independent of the atom chains, with $(W_{t_{k+1}}-W_{t_k})/\sqrt h$ standard normal. The fine chain is unchanged by the construction, and its own uniforms $(U'_j)_{j\ge 2k}$ are independent of $\cG _k$ .∎

Lemma 4.3 (The whole hierarchy on one space) . Let $\mathsf r$ be a scheme. There is a probability space carrying chains $(y^{(m)},X^{(m)})_{m\ge 0}$ , each with the law of its resolution, such that every neighbouring pair is coupled as in Definition 4.1. Writing $\cG _t:=\sigma \bigl (\bigcup _m\cF ^{(m)}_t\bigr )$ for the filtration generated by all of them, every $y^{(m)}$ is co-immersed in $(\cG _t)$ : for $t_k$ on its grid, $\Q \bigl (y^{(m)}_{k+1}\in \cdot \mid \cG _{t_k}\bigr )=P_m\bigl (y^{(m)}_k,\cdot \bigr )$ , and $W$ is a $(\cG _t)$ -Brownian motion.

Proof.For each $M$ build $(y^{(0)},\dots ,y^{(M)})$ downwards from $y^{(M)}$ by Definition 4.1, which expresses $y^{(m)}$ as a measurable function of $y^{(m+1)}$ and an independent uniform array. By Lemma 4.2 each $y^{(m)}$ so built has the law of its resolution, so the law of $(y^{(0)},\dots ,y^{(M)})$ does not depend on $M$ : the family of finite-dimensional laws is projective, and Kolmogorov’s extension theorem gives one space carrying the whole sequence, with $W$ adjoined independently of the atom arrays. For the co-immersion, fix $M\ge m$ . The vector $(y^{(0)},\dots ,y^{(M)})$ is a Markov chain in its own filtration, because each of its coordinates moves by a function of the current states and of fresh uniforms; hence the conditional law of $y^{(m)}_{k+1}$ given the past of the whole vector is $P_m(y^{(m)}_k,\cdot )$ , by Lemma 4.2 applied inside that vector. This holds for every $M$ , and $\cG _{t_k}$ is generated by the increasing union of those pasts, so the martingale convergence theorem passes the identity to the limit. The same argument gives the Brownian property of $W$ , whose increments after $t_k$ are independent of every $\cF ^{(m)}_{t_k}$ .∎

4.2The refinement bound

Lemma 4.4 (Refinement bound) . Let $r=r_m$ and $r'=r_{m+1}$ be neighbouring resolutions of a scheme of order $\rho$ , coupled as in Definition 4.1, and put $e_k:=|y_k-y'_{2k}|$ . Then

\begin{equation} \bigl \|e_k\bigr \|_{L^\rho }\ \le \ \varepsilon _m\qquad \text {for every }k\le n, \label {eq:refinement-bound} \end{equation}

with

\varepsilon _m

the refinement modulus (14) of Definition 2.6, and consequently

\|\nu _{y_k}-\nu _{y'_{2k}}\|_{L^{\rho /2}}\le \varepsilon _m\|y_k+y'_{2k}\|_{L^\rho }

. The three parts of the numerator of (14) appear in the proof as the three things that can go wrong in one step — the defect below the level, the projection of the coarse atom into the finer grid, and the excursion above the level — while

\iota _{\rho ,m}

is the initial projection. The excursion is not an afterthought: it enters at every step and is amplified by the same geometric sum as the other two, which is why (S4) constrains the level rather than merely asserting that the tail is small.

Proof.Condition on $\cG _k$ and let $\hat y_k:=\pi '(y_k)$ , so $|\hat y_k-y_k|\le \zeta _m$ by (9). By construction $y_{k+1}$ and $y'_{2k+2}$ are the two quantile functions of $P(y_k,\cdot )$ and $(P')^2(y'_{2k},\cdot )$ evaluated at the same uniform level, which is the quantile coupling; since that coupling attains $W_\rho$ in one dimension,

\begin{align*} \E \bigl [e_{k+1}^\rho \mid \cG _k\bigr ]^{1/\rho } &=W_\rho \bigl (P(y_k,\cdot ),(P')^2(y'_{2k},\cdot )\bigr )\\ &\le W_\rho \bigl (P(y_k,\cdot ),(P')^2(\hat y_k,\cdot )\bigr ) +W_\rho \bigl ((P')^2(\hat y_k,\cdot ),(P')^2(y'_{2k},\cdot )\bigr ) \end{align*}

by the triangle inequality. On $\{y_k\le \ell _m\}$ the first term is at most $d_{\ell _m}(r,r')$ by (10); the second is at most $(c')^2|\hat y_k-y'_{2k}|\le (c')^2(e_k+\zeta _m)$ , because (C1) for $r'$ gives the factor $c'$ for one step of $P'$ and hence $(c')^2$ for two, by the triangle inequality along the grid and composition. On $\{y_k>\ell _m\}$ use the crude bound $W_\rho \le \Xi _\cA$ . Taking $L^\rho$ norms and using Minkowski’s inequality, ∎

\|e_{k+1}\|_{L^\rho }\le (c')^2\|e_k\|_{L^\rho }+d_{\ell _m}+\zeta _m +\Xi _\cA \,p_{\ell _m}^{1/\rho },

where

p_{\ell _m}:=\max _{k<n}\Q (y_k>\ell _m)\le C_0e^{-\beta \ell _m^2}

by Lemma 5.1 and

\Xi _\cA :=\max \cA \vee \max \cA '=\Xi _m

by (S1), so the last term is at most the charge

\chi _m

of (12). By (S2),

y_0=\pi (y'_0)

has the coarse initial law and

\|e_0\|_{L^\rho }=\iota _{\rho ,m}

. Iteration gives the geometric contribution

(d_{\ell _m}+\zeta _m+\chi _m)/(1-(c')^2)

, while the initial error survives as at most

\iota _{\rho ,m}

, which is (14). The last statement is

|\nu -\nu '|=|y-y'|(y+y')

with Hölder’s inequality.

The four ingredients of $\varepsilon _m$ are exactly the computed quantities of Definition 2.6: the initial and inter-grid projection errors, the truncated defect, the excursion charge, and the contraction budget $1-c_{m+1}^2$ . The bound is in $L^\rho$ rather than pathwise, and this is not a technical preference. A pathwise bound would need (C1) in the uniform metric $W_\infty$ , and a quantized kernel does not contract there: two adjacent atoms’ quantile functions differ by a full atom on some level, so the uniform factor is $1$ however fine the mesh, and slacks add along the grid instead of composing (Remark 2.5). What is genuinely absent here, and unavoidable against a continuous target, is an excursion term for the target: both objects live on finite atom sets, so the only tail charge is the one displayed, and (S4) is exactly what keeps it at the grid scale.

4.3Transfer between resolutions

Proposition 4.5 (Transfer of conditional readouts at common dates) . Let $\rho \ge 1$ and let $Y,Y'$ be integrable variables measurable for the coarse and the fine path respectively, with $R_k:=\E [Y\mid \cF _k]$ and $R'_k:=\E [Y'\mid \cF '_{2k}]$ . Then

\begin{equation} \Bigl \|\max _{k\le n}\bigl |R_k-R'_k\bigr |\Bigr \|_{L^\rho } \ \le \ d_\rho \,\bigl \|Y-Y'\bigr \|_{L^\rho }, \qquad d_\rho :=\rho /(\rho -1)\ \ (\rho >1), \label {eq:grid-transfer} \end{equation}

and for

\rho =1

the same holds with

\max _k

replaced by any fixed

k

Proof.By Lemma 4.2 both $R_k$ and $R'_k$ are conditional expectations with respect to the same filtration $(\cG _k)$ , so $R_k-R'_k=\E [Y-Y'\mid \cG _k]$ is a martingale in $k$ closed by $Y-Y'$ ; apply Doob’s inequality.∎

This is where the internal comparison is simpler than a comparison with a model. There the two conditional expectations lived in different filtrations, and the estimate needed the chain to be co-immersed in the target, which held at grid dates and failed between them. Here both readouts are conditional expectations in one filtration by construction, and the only thing left to bound is the payoff difference $Y-Y'$ , which Lemma 4.4 controls in $L^\rho$ .

Remark 4.6 (Between common dates) . The coarse chain is constant on $[t_k,t_{k+1})$ while the finer one moves once inside it. A uniform-in-time comparison therefore carries, besides (23), the finer chain’s own oscillation over a coarse cell, which is a readout of the finer resolution and is estimated in Section 5 from (C2)–(C4). This is the same two-term structure as in a comparison with a continuous model, with the finer chain in the role the model used to play; the difference is that both terms are now computable from the objects at hand.

5Refinement rates and the limit

Two resolutions of a scheme now differ by (22) in the volatility coordinate. This section turns that into rates for the readouts, and then lets the resolution go to zero: the family is Cauchy, its limit is a continuous market carrying the same rows, and every readout covered below is within an explicit distance of its limit. No object outside the scheme appears anywhere.

5.1Moduli of one resolution

Lemma 5.1 (Uniform tails from (C2) ) . Let $r$ satisfy (C2)–(C3). Then for $x\ge 0$

\begin{equation} \Q \Bigl (\max _{k\le n}\nu _{y_k}\ge x\Bigr )\ \le \ C_0\,e^{-\beta x}, \qquad C_0:=e^{c_\beta H}\,\E _{\mu _0}V_\beta , \label {eq:tail} \end{equation}

and consequently

\bigl \|\max _k\nu _{y_k}\bigr \|_{L^\rho }\le m_\rho <\infty

and

\bigl \|\max _ky_k\bigr \|_{L^\rho }\le m_\rho ^{1/2}

, with

m_\rho

depending only on

\beta

C_0

and

\rho

. For a scheme these are uniform in

m

c_\beta

and

\beta

by (S3), and

C_0=e^{c_\beta H}\sup _m\E _{\mu _{0,m}}V_\beta <\infty

by (S2), which is where uniform compatible initial control is used — finiteness of

\E _{\mu _{0,m}}V_\beta

at each

m

separately would leave

C_0

unbounded.

Proof.By (C2) the process $M_k:=e^{-c_\beta t_k}V_\beta (y_k)$ is a nonnegative supermartingale, so Doob’s maximal inequality gives $\Q (\max _kM_k\ge \ell )\le \E M_0/\ell$ . Since $V_\beta (y_k)\le e^{c_\beta H}M_k$ for $k\le n$ , taking $\ell =e^{-c_\beta H}e^{\beta x}$ yields (24); integrating the tail gives the moments.∎

Lemma 5.2 (Cell moves and their maxima from (C4) ) . Let $r$ satisfy (C4) and put

\begin{equation} \Lambda _n:=\sqrt {h}\,\bigl (1+\sqrt {\log (2n)}\bigr ). \label {eq:Lambda} \end{equation}

Then for every

\rho \ge 1

(a): $\bigl \|\,\max _{k<n}|y_{k+1}-y_k|\,\bigr \|_{L^\rho }\le C^y_\rho \,\vartheta ^{-1/2}\Lambda _n$ ;
(b): $\bigl \|\,\max _{k<n}|\nu _{y_{k+1}}-\nu _{y_k}|\,\bigr \|_{L^\rho }\le C^\nu _\rho \,\vartheta ^{-1/2}\Lambda _n$ , by Cauchy–Schwarz with Lemma 5.1;
(c): the interpolated log-spot satisfies $\bigl \|\max _{k<n}\sup _{t\in [t_k,t_{k+1}]}|X_t-X_{t_k}|\bigr \|_{L^\rho }\le C^X_\rho \,\Lambda _n$ ,

with $C^y_\rho$ , $C^\nu _\rho$ , $C^X_\rho$ depending only on $\rho$ , $\beta$ and $C_0$ .

Proof.(a) By (C4) and Markov’s inequality, $\Q (|y_{k+1}-y_k|>s\sqrt {h/\vartheta }\mid \cF _k)\le 2e^{-s^2}$ , so a union bound over the $n$ steps gives $\Q (\max _k|y_{k+1}-y_k|>s\sqrt {h/\vartheta })\le 2ne^{-s^2}$ ; integrating $\rho s^{\rho -1}$ against this tail from $s_0=\sqrt {\log (2n)}$ upwards, and bounding the contribution below $s_0$ by $s_0^\rho$ , gives (a). (b) follows from $|\nu _{y_{k+1}}-\nu _{y_k}|=|y_{k+1}-y_k|(y_{k+1}+y_k)$ , Cauchy–Schwarz and Lemma 5.1. (c) On a cell $|X_t-X_{t_k}|\le \tfrac 12h\nu _{y_k}+\sqrt {\nu _{y_k}}\sup _{t}|W_t-W_{t_k}|$ , and the maximum over cells of the Brownian oscillation has the same $\sqrt h\sqrt {\log 2n}$ order by the reflection principle and a union bound; multiply by $\max _ky_k$ and use Lemma 5.1.∎

5.2Rates between two resolutions

Theorem 5.3 (Refinement of budgets, index and index prices) . Let $\rho >1$ , let $\mathsf r$ be a scheme of order $4\rho$ — the order at which Hölder’s inequality reads the modulus below — and let $r=r_m$ , $r'=r_{m+1}$ be coupled as in Definition 4.1. For every grid-aligned window $(u,w)$ with $u,w$ on the coarse grid,

\begin{equation} \Bigl \|\max _{k:\,t_k\le u}\bigl |B(t_k;u,w)-B'(t_k;u,w)\bigr |\Bigr \|_{L^\rho } \le d_\rho \,(w-u)\,\Bigl (2\,\varepsilon ^{(4\rho )}_m\, \bigl \|\max _{0\le k\le n}(y_k+y'_{2k})\bigr \|_{L^{4\rho }} +C^\nu _{\rho }\vartheta ^{-1/2}\Lambda _{2n}\Bigr ). \label {eq:cir-budget-rate} \end{equation}

The rolling index, whose window moves with the valuation date and which therefore is not covered by any single application of the display above, obeys

\begin{equation} \Bigl \|\max _{k:\,t_k\le T_{\rm op}}\bigl |U_{t_k}-U'_{t_k}\bigr |\Bigr \|_{L^\rho } \le d_\rho \,\frac H\Delta \,\Bigl (2\,\varepsilon ^{(4\rho )}_m\, \bigl \|\max _{0\le k\le n}(y_k+y'_{2k})\bigr \|_{L^{4\rho }} +C^\nu _{\rho }\vartheta ^{-1/2}\Lambda _{2n}\Bigr ). \label {eq:cir-rolling-rate} \end{equation}

If the two index maps have the certified common floor

u_-:=\min _iU_i\wedge \min _iU'_i>0

, the index inherits the bound:

\begin{equation} \Bigl \|\max _k\bigl |V_{t_k}-V'_{t_k}\bigr |\Bigr \|_{L^\rho } \le \frac 1{2\sqrt {u_-}}\,\Bigl \|\max _k\bigl |U_{t_k}-U'_{t_k}\bigr |\Bigr \|_{L^\rho }, \label {eq:cir-vix-price-rate} \end{equation}

and the unconditional price of every

L_g

-Lipschitz payoff of the index at a grid fixing differs by at most

L_g

times the right-hand side of (28). By Proposition 4.5, the maximum difference of the corresponding conditional-price processes at common dates is at most

d_\rho L_g

times that same right-hand side. Such a floor is an additional family certificate, not a consequence of a positive minimum at each finite resolution. Without it the index still obeys the square-root form

\|\max _k|V_{t_k}-V'_{t_k}|\|_{L^\rho }\le \|\max _k|U_{t_k}-U'_{t_k}|\|_{L^{\rho /2}}^{1/2}

, which needs no floor. Section 8 computes the finite-level floors of the displayed square-root family; the exact CIR mean formula cannot be substituted for the quantized matrix without a separate perturbation bound.

Proof.Apply Proposition 4.5 with $Y:=A_w-A_u$ and $Y':=A'_w-A'_u$ . Over one coarse step $[t_i,t_{i+1}]$ inside the window the coarse clock gains $h\nu _{y_i}$ and the fine clock $\tfrac h2(\nu _{y'_{2i}}+\nu _{y'_{2i+1}})$ , so their difference is at most $\tfrac h2\bigl (2|\nu _{y_i}-\nu _{y'_{2i}}|+|\nu _{y'_{2i+1}}-\nu _{y'_{2i}}|\bigr )$ . By Hölder’s inequality with the two exponents $4\rho$ and Lemma 4.4 at that order, $\|\nu _{y_i}-\nu _{y'_{2i}}\|_{L^{2\rho }}\le \varepsilon ^{(4\rho )}_m\|y_i+y'_{2i}\|_{L^{4\rho }}$ , where $\varepsilon ^{(4\rho )}_m$ is the modulus (22) read at that order; the last term is the finer chain’s own one-step move, bounded in $L^\rho$ by Lemma 5.2 (b) at resolution $r'$ . Summing over the $(w-u)/h$ steps of the window and taking norms gives (26). For (27) the window is not fixed, so Doob’s inequality cannot be applied to a martingale closed by $A_w-A_u$ . Instead write $U_{t_k}-U'_{t_k}=\Delta ^{-1}\E [Z_k-Z'_k\mid \cG _{t_k}]$ with $Z_k:=A_{t_k+\Delta }-A_{t_k}$ , and dominate every window by the horizon: $\max _k|Z_k-Z'_k|\le D:=\sum _{i<n}h\bigl (2|\nu _{y_i}-\nu _{y'_{2i}}|+|\nu _{y'_{2i+1}}-\nu _{y'_{2i}}|\bigr )$ , one single random variable. Then $\max _k|U_{t_k}-U'_{t_k}|\le \Delta ^{-1}\max _k\E [D\mid \cG _{t_k}]$ , Doob’s inequality applies to the martingale $\E [D\mid \cG _{t_k}]$ , and $\|D\|_{L^\rho }$ is bounded by the same per-step estimates summed over the $n$ steps of the horizon, which is the factor $H/\Delta$ after dividing by $\Delta$ . For (28), $U\ge u_-$ and $U'\ge u_-$ because every entry of $P^m\nu$ is a convex combination of labels, so $|\sqrt U-\sqrt {U'}|\le |U-U'|/(2\sqrt {u_-})$ .∎

Corollary 5.4 (Refinement of log-spot and of spot prices) . Under the hypotheses of Theorem 5.3, for $\rho \ge 2$ ,

\begin{equation} \bigl \|X_{t_n}-X'_{t_n}\bigr \|_{L^\rho } \le \tfrac 12\bigl \|A_{t_n}-A'_{t_n}\bigr \|_{L^\rho } +C^{\rm BDG}_\rho \sqrt H\,\Bigl (\varepsilon ^{(\rho )}_m+C^y_{2\rho }\vartheta ^{-1/2}\Lambda _{2n}\Bigr ). \label {eq:cir-spot-error} \end{equation}

Consequently every bounded

L_g

-Lipschitz payoff of the log-spot at the horizon has a price difference at most

L_g

times the left-hand side. The displayed bound is

O(\varepsilon ^{(4\rho )}_m+\Lambda _{2n})

under the theorem’s hypotheses.

There is also a genuine spot-payoff bound. Let $p,q>1$ be conjugate exponents with $2\le q\le \rho$ , and put $a_p:=\tfrac 12p(p-1)H$ . If $a_p<\beta$ , then for every $L_g$ -Lipschitz payoff $g$ of $S_{t_n}$ ,

\begin{equation} \bigl |\E g(S_{t_n})-\E g(S'_{t_n})\bigr | \le L_g C_p\,\|X_{t_n}-X'_{t_n}\|_{L^q}, \qquad C_p:=2e^{x_0}\left (1+\frac {a_pC_0}{\beta -a_p}\right )^{1/p}. \label {eq:cir-spot-price-error} \end{equation}

The

L^q

term has the same refinement order, read at order

4q

, which is available from the order-

4\rho

scheme by Remark 2.8.

Proof.The two log-spots share the Brownian motion, so $X_{t_n}-X'_{t_n}=-\tfrac 12(A_{t_n}-A'_{t_n})+\int _0^H(\sqrt {\nu _{y_{k(s)}}}-\sqrt {\nu _{y'_{j(s)}}})\,dW_s$ with $k(s)$ , $j(s)$ the coarse and fine indices. The integrand is at most $|y_{k(s)}-y'_{2k(s)}|+|y'_{2k(s)}-y'_{j(s)}|$ , the first bounded in $L^\rho$ by $\varepsilon ^{(\rho )}_m$ and the second by the finer chain’s one-step move; apply the Burkholder–Davis–Gundy inequality and Lemma 5.2 (a). The clock difference is bounded as in Theorem 5.3. For the spot-payoff statement, $|e^x-e^{x'}|\le (e^x+e^{x'})|x-x'|$ and Hölder’s inequality give the first bound in (30). Moreover ∎

\|S_{t_n}\|_{L^p}^p =e^{px_0}\E \exp \!\left (\tfrac 12p(p-1)A_{t_n}\right ) \le e^{px_0}\left (1+\frac {a_pC_0}{\beta -a_p}\right ),

because

A_{t_n}\le H\max _k\nu _{y_k}

and (24) integrates at every rate

a_p<\beta

; the same bound holds for

S'

. Apply the log-spot estimate at order

q

to finish.

Proposition 5.5 (Refinement of the physical layer) . Let $L,L'$ be the densities (5) of the two resolutions and $\Pp =L\Q$ , $\Pp '=L'\Q$ on the coupled space. Then $R:=L/L'=\mathcal E\bigl (\lambda \int _0^\cdot (\sqrt {\nu _{y_{k(s)}}}-\sqrt {\nu _{y'_{j(s)}}})\,dW^{\Pp '}_s\bigr )_H$ ,

\begin{equation} H(\Pp \,|\,\Pp ')=\tfrac 12\lambda ^2\,\E _{\Pp }\!\int _0^H\!\bigl (y_{k(s)}-y'_{j(s)}\bigr )^2ds, \qquad \|\Pp -\Pp '\|_{\rm TV}\le \bigl (\tfrac 12H(\Pp \,|\,\Pp ')\bigr )^{1/2}, \label {eq:cir-entropy} \end{equation}

and for every grid date and every

f

bounded by

M

and

L_f

-Lipschitz,

\begin{equation} \E _{\Pp }\bigl |\E _{\Pp }[f(X_{t_n})\mid \cF _{k}]-\E _{\Pp '}[f(X'_{t_n})\mid \cF '_{2k}]\bigr | \le L_f\,\E _{\Pp }\bigl |X_{t_n}-X'_{t_n}\bigr | +2M|\lambda |\Bigl (\E _\Pp \!\int _0^H\!(y_{k(s)}-y'_{j(s)})^2ds\Bigr )^{1/2}, \label {eq:cir-l1-physical} \end{equation}

which is

O(\varepsilon ^{(2)}_m+\Lambda _{2n})

with no condition on

\lambda

Proof.Both densities are stochastic exponentials of integrals against the same $W$ with integrands measurable for the atom chains, so $R$ is the stated exponential after the substitution $dW=dW^{\Pp '}+\lambda y'\,ds$ , and its $\Pp$ -expected logarithm is the entropy in (31); Pinsker’s inequality gives the total variation bound. For (32), first note that the atom arrays keep their $\Q$ -law under the respective physical measures and the shifted Brownian motion remains independent of them. Thus the co-immersion identities put both conditional readouts in the common filtration $\cG _k$ . Split the difference there into the payoff term, whose $\Pp$ -expectation is at most $L_f\E _{\Pp }|X_{t_n}-X'_{t_n}|$ , and the change-of-measure term. If $l_k:=\E [L\mid \cG _k]$ , $l'_k:=\E [L'\mid \cG _k]$ and $|Z|\le M$ , then adding and subtracting the two conditional numerators gives ∎

\E _\Q \!\left [l_k\left | \frac {\E [LZ\mid \cG _k]}{l_k}- \frac {\E [L'Z\mid \cG _k]}{l'_k}\right |\right ] \le 2M\E _\Q |L-L'|=4M\|\Pp -\Pp '\|_{\rm TV}.

Together with (31), this is the second term displayed in (32); its constant is unchanged. For the order, both expectations in (32) are under

\Pp

, and they are nevertheless the ones already bounded. Let

\cA _\infty

be the

\sigma

-field generated by the two atom paths and the randomizations of Definition 4.1. The normals

(\xi _k)

are independent of

\cA _\infty

and

\E [L\mid \cA _\infty ]=1

by Theorem 3.1 (d), so

\Pp

and

\Q

agree on

\cA _\infty

: any

\cA _\infty

-measurable expectation may be read under

\Q

. The second integrand is

\cA _\infty

-measurable, so its

\Pp

-expectation is its

\Q

-expectation, which Lemma 4.4 at order

2

and Lemma 5.2 (a) bound by

(\varepsilon ^{(2)}_m+C\Lambda _{2n})^2

times

H

. For the first,

W^\Pp :=W-\lambda \int y\,ds

is a

\Pp

-Brownian motion and

X_{t_n}-X'_{t_n}=-\tfrac 12\bigl (A_{t_n}-A'_{t_n}\bigr ) +\lambda \!\int _0^H\!(y-y')y\,ds+\int _0^H\!(y-y')\,dW^\Pp ,

whose first two terms are

\cA _\infty

-measurable and whose third has

\Pp

-expected modulus at most

C\,\E _\Pp (\int _0^H(y-y')^2ds)^{1/2}

by the Burkholder–Davis–Gundy inequality, again an

\cA _\infty

-expectation. No integrability of

L

is used, which is why no band on

\lambda

appears.

Nothing here needs a band on $\lambda$ , and the reason is structural rather than quantitative. In a comparison with a continuous model the band paid for the model’s excursions above the chain’s cap. Between two resolutions there is no such model: the two coupling errors are functionals of the atom paths, the change of measure acts on the return noise alone and leaves those paths’ joint law alone, so every quantity in (31) and (32) is one already bounded under $\Q$ . What survives is the excursion charged in (22), made negligible by the choice of level.

5.3The limit of a scheme

For this subsection, extend every resolution to continuous time by setting $y^{(m)}_t:=y^{(m)}_k$ on $[t^{(m)}_k,t^{(m)}_{k+1})$ , $y^{(m)}_H:=y^{(m)}_{n_m}$ , and

\begin{equation} A^{(m)}_t:=\int _0^t\bigl (y^{(m)}_s\bigr )^2ds . \label {eq:clock-continuous-time} \end{equation}

Thus (33) agrees with (3) at grid dates. On the same half-open cell let $\cF ^{(m),0}_t:=\cF ^{(m)}_k\vee \sigma (W_s:s\le t)$ , put $\cG ^0_t:=\sigma (\bigcup _m\cF ^{(m),0}_t)$ , and let $(\cG _t)$ denote the $\Q$ -completed, right-continuous augmentation of $(\cG ^0_t)$ . The Brownian and co-immersion conclusions of Lemma 4.3 are unchanged by this usual augmentation. In particular, the conditional-expectation martingales below admit right-continuous versions and Doob’s continuous-time inequality applies.

Theorem 5.6 (Existence of the limit and convergence of the readouts) . Let $\rho >2$ , let $\mathsf r$ be a scheme of order $4\rho$ — the order at which Theorem 5.3 reads the modulus, and one that Lemma 2.7 supplies — with $\sum _mn_m^{1/\rho }\bigl (\varepsilon ^{(4\rho )}_m+\Lambda _{2n_m}\bigr )<\infty$ , and realize all its resolutions on one space as in Lemma 4.3. For the uniform-time assertion, extend the grid-date index by

\begin{equation} U^{(m)}_t:=\frac 1\Delta \E \!\left [A^{(m)}_{t+\Delta }-A^{(m)}_t\mid \cG _t\right ], \qquad 0\le t\le T_{\rm op}; \label {eq:index-continuous-time} \end{equation}

co-immersion makes this agree with (4) at every grid date. Then:

(a): the piecewise-constant volatility paths converge uniformly on $[0,H]$ , almost surely, to a process $y^\infty$ with continuous paths, with $\bigl \|\sup _t|y^{(m)}_t-y^\infty _t|\bigr \|_{L^\rho }\le \sum _{j\ge m}\bigl (n_j^{1/\rho }\varepsilon ^{(\rho )}_j+\|J_{j+1}\|_{L^\rho }\bigr )$ , $J_j$ being the largest one-step move of resolution $j$ ;
(b): the clocks converge uniformly to a continuous increasing $A^\infty _t=\int _0^t(y^\infty _s)^2ds$ , and the log-spots converge uniformly in $L^\rho$ to a continuous $X^\infty$ with $X^\infty _t=x_0-\tfrac 12A^\infty _t+\int _0^ty^\infty _s\,dW_s$ ;
(c): in the augmented joint filtration just defined, $W$ is a Brownian motion and every $y^{(m)}$ , hence $y^\infty$ , is adapted. Then $S^\infty =e^{X^\infty }$ is a $(\cG _t)$ -martingale, the log-contract identity $\E [X^\infty _w-X^\infty _u\mid \cG _t]=-\tfrac 12B^\infty (t;u,w)$ holds with $B^\infty (t;u,w):=\E [A^\infty _w-A^\infty _u\mid \cG _t]$ , that field is nonnegative, additive in the window and a martingale in its valuation date, and the chains’ indices converge to the limit’s: $\sup _{t\le T_{\rm op}}|U^{(m)}_t-U^\infty _t|\to 0$ in $L^\rho$ with $U^\infty _t:=\Delta ^{-1}B^\infty (t;t,t+\Delta )$ ;
(d): the physical layer passes to the limit: the densities converge, $L^{(m)}\to L^\infty :=\exp \bigl (\lambda \int _0^Hy^\infty _sdW_s-\tfrac 12\lambda ^2A^\infty _H\bigr )$ in $L^1(\Q )$ , with $\E L^\infty =1$ and no condition on $\lambda$ , so $\Pp ^{(m)}\to \Pp ^\infty :=L^\infty \Q$ in total variation, the density process is $D^\infty _t=\exp (\lambda \int _0^ty^\infty dW-\tfrac 12\lambda ^2A^\infty _t)$ , the atom-path law is the same under $\Pp ^\infty$ as under $\Q$ , and every Bayes row of Theorem 3.1(d) holds in the limit, and for every bounded $L_f$ -Lipschitz $f$ and every grid date the physical conditional readouts converge,
$\begin{equation} \begin {split} \E _{\Pp ^{(m)}}\bigl |\E _{\Pp ^{(m)}}&[f(X^{(m)}_H)\mid \cG _t]-\E _{\Pp ^\infty }[f(X^\infty _H)\mid \cG _t]\bigr |\\ &\le \ L_f\,\E _{\Pp ^{(m)}}\bigl |X^{(m)}_H-X^\infty _H\bigr | +2\|f\|_\infty \,\E \bigl |L^{(m)}-L^\infty \bigr |\ \longrightarrow \ 0 ; \end {split} \label {eq:limit-physical} \end{equation}$
(e): each risk-neutral readout covered by Proposition 4.5, Theorem 5.3 or Corollary 5.4 is within the sum over $j\ge m$ of its applicable two-resolution bounds. The bounded Lipschitz physical log-spot readouts of (d) obey the direct common-space estimate (35); their rate is obtained from the tails of $\|y^{(m)}-y^\infty \|$ and $\|A^{(m)}-A^\infty \|$ , rather than by telescoping expectations taken under different physical measures.

Proof.(a): on the coupled space, for $t\in [t_k,t_{k+1})$ of the coarse grid, $|y^{(m)}_t-y^{(m+1)}_t|\le |y^{(m)}_{t_k}-y^{(m+1)}_{t_k}|+|y^{(m+1)}_{t_k}-y^{(m+1)}_t|$ , so $\sup _t|y^{(m)}_t-y^{(m+1)}_t|\le \max _{k\le n_m}e_k+J_{m+1}$ . The maximum over the $n_m$ dates is controlled from the pointwise bounds of Lemma 4.4 by $\|\max _ke_k\|_{L^\rho }\le n_m^{1/\rho }\max _k\|e_k\|_{L^\rho }\le n_m^{1/\rho }\varepsilon ^{(\rho )}_m$ , and $\|J_j\|_{L^\rho }\le C\Lambda _{2n_j}$ by Lemma 5.2 (a); the hypothesis makes the sum of these finite, so the sequence is uniformly Cauchy almost surely by the Borel–Cantelli lemma. The limit is continuous as a uniform limit of piecewise-constant paths whose jumps tend to zero. (b): uniform convergence and the exponential moment of Lemma 5.1 give clock convergence in $L^\rho$ by Hölder and interpolation. BDG at order $\rho$ then gives uniform $L^\rho$ convergence of the stochastic integrals, with the same $W$ throughout. (c): $W$ is a Brownian motion in $(\cG _t)$ , being independent of the uniforms that drive every atom chain, and $y^\infty$ is $\cG$ -adapted by (a), so $\int y^\infty dW$ is a $\cG$ -martingale and $S^\infty =\mathcal E(\int y^\infty dW)$ is a $\cG$ -local martingale; it is a true martingale because $\E e^{pX^\infty _H}=e^{px_0}\E \exp (\tfrac 12p(p-1)A^\infty _H)<\infty$ for $p>1$ close to $1$ , by (24) and $A^\infty _H\le H\sup _t(y^\infty _t)^2$ , which gives uniform integrability. The log-contract identity is then immediate from $X^\infty =x_0-\tfrac 12A^\infty +\int y^\infty dW$ and the definition of $B^\infty$ , and additivity and the martingale property in the valuation date are properties of that conditional expectation. For the indices, the filtrations $(\cF ^{(m)})$ are not nested, the couplings being randomized, so no interchange of limits is attempted: instead Definition (34) puts every resolution in the filtration of the whole hierarchy, and conditional expectation is an $L^\rho$ contraction. Uniformity in $t$ is not automatic there either, the window moving with $t$ ; as in Theorem 5.3 dominate every window by the horizon, $\sup _{t\le T_{\rm op}}|U^{(m)}_t-U^\infty _t|\le \Delta ^{-1}\sup _t\E [D^{(m)}\mid \cG _t]$ with the single variable $D^{(m)}:=\int _0^H|(y^{(m)}_s)^2-(y^\infty _s)^2|ds$ , and apply Doob’s inequality to that martingale; Hölder and interpolation between (a) and the uniform exponential moments give $\|D^{(m)}\|_{L^\rho }\to 0$ . (d): the exponents converge in $L^2$ by (a) and Itô’s isometry, so $L^{(m)}\to L^\infty$ in probability. Both the chains’ volatilities and $y^\infty$ are measurable for the $\sigma$ -field $\cA _\infty$ generated by the atom arrays, which is independent of $W$ ; conditionally on it the exponent of $L^\infty$ is Gaussian with mean $-\tfrac 12\lambda ^2A^\infty _H$ and variance $\lambda ^2A^\infty _H$ , so $\E [L^\infty \mid \cA _\infty ]=1$ and $\E L^\infty =1$ — with no integrability condition on $\lambda$ , in contrast with the Novikov route, which would ask $\lambda ^2H<2\beta$ . Since $L^{(m)}\ge 0$ , $\E L^{(m)}=1=\E L^\infty$ and $L^{(m)}\to L^\infty$ in probability, Scheffé’s lemma gives $L^{(m)}\to L^\infty$ in $L^1$ , hence $\|\Pp ^{(m)}-\Pp ^\infty \|_{\rm TV}\le \tfrac 12\E |L^{(m)}-L^\infty |\to 0$ ; the remaining rows are the identities of Theorem 3.1 (d) read for $L^\infty$ , the conditioning on $\cA _\infty$ leaving the atom-path law unchanged. For (35), conditional readouts under two different measures are not compared by total variation alone: write both as $\Q$ -conditional expectations, $\E _{\Pp }[f\mid \cG _t]=\E [L_{t,H}f\mid \cG _t]$ with $L_{t,H}$ the density over $(t,H]$ , and split the difference into a payoff term, bounded by $L_f\E _{\Pp ^{(m)}}|X^{(m)}_H-X^\infty _H|$ , and a density term, bounded by $2\|f\|_\infty \E |L^{(m)}-L^\infty |$ . Both terms have direct quantitative bounds. The density-ratio entropy calculation of Proposition 5.5 gives ∎

\E |L^{(m)}-L^\infty | \le |\lambda |\Bigl (\E _\Q \int _0^H|y^{(m)}_s-y^\infty _s|^2ds\Bigr )^{1/2}.

Under

\Pp ^{(m)}

W^{(m)}=W-\lambda \int y^{(m)}ds

is Brownian, and

X^{(m)}_H-X^\infty _H=-\tfrac 12(A^{(m)}_H-A^\infty _H) +\lambda \int _0^H(y^{(m)}-y^\infty )y^{(m)}ds +\int _0^H(y^{(m)}-y^\infty )dW^{(m)}.

Every coefficient outside the last Brownian integral is atom-measurable, so its

\Pp ^{(m)}

expectation equals its

\Q

expectation; BDG bounds the last term. The tails in (a) therefore give the physical rate directly. Finally, (e) is the triangle inequality along the scheme for risk-neutral readouts and these direct bounds for physical ones.

Remark 5.7 (What the limit theorem replaces) . A statement that a finite model approximates a given continuous market must name that market and inherit its hypotheses. Theorem 5.6 says instead that the family a modeller actually builds determines its own continuum object and comes with a computable distance to it. The continuous market is an output, and by construction it carries the structural rows, so it is a legitimate target for the certification of Huang [29] rather than an assumption imported from outside.

6What revelation costs

The estimates of Sections 4 and 5 compare conditional readouts date by date in one filtration. Primitive path laws alone cannot do this job. A chain and its revealed twin have the same law for $(y,X)$ , and hence the same unconditional prices for bounded payoffs measurable from that path, but their rolling-index gap is kernel-computable and converges to a limit defect. That limit is nonzero unless the forward window clock is already measurable at the valuation date.

The enlargement below is an initial enlargement by the whole atom path, of the kind studied by Jacod [31], and the phenomenon it exposes is the one separating the mixture diffusion of Brigo and Mercurio [14] from the uncertain-volatility model carrying the same marginals [13], in which a volatility drawn at the initial date leaves every vanilla price there intact and changes every conditional law after it. What a resolution supplies is that separation as a number: (39) computes it from $(P,\nu )$ , and Proposition 6.3 identifies its limit.

Definition 6.1 (The revealed chain) . Let $r$ be a resolution with chain $(y,X)$ and filtration $(\cF _k)$ . The revealed chain is the same pair carried by the enlarged filtration

\begin{equation} \cG _k:=\cF _k\vee \sigma \bigl (y_i:\ i\le n\bigr ), \label {eq:revealed-filtration} \end{equation}

in which the whole atom path is known at time zero. Its readouts are those of Definition 2.3 computed in

(\cG _k)

Proposition 6.2 (Premature revelation) . Let $r$ be a resolution. Then, under $\Q$ :

(a): the revealed chain has the same law as the chain and $S=e^X$ is a $(\cG _k)$ -martingale, so it is again a market carrying the same traded assets with the same joint law; every marginal of $(y,X)$ , and every unconditional expectation $\E f(y,X)$ of a bounded payoff, is unchanged. Its conditional prices are not: $\cG _0$ already knows the atom path;
(b): its budget field is the realized clock, $B^{\rm rev}(t_k;u,w)=A_w-A_u$ , so its index is the realized window average
$\begin{equation} U^{\rm rev}_{t_k}=\frac h\Delta \sum _{m<m_\Delta }\nu _{y_{k+m}}, \qquad \text {while}\qquad U_{t_k}=\E \bigl [U^{\rm rev}_{t_k}\mid \cF _k\bigr ]; \label {eq:revealed-index} \end{equation}$
(c): the two differ by the revelation defect
$\begin{equation} \cR (t_k):=\bigl \|U^{\rm rev}_{t_k}-U_{t_k}\bigr \|_{L^2}=\bigl (\E \,w(y_k)\bigr )^{1/2}, \qquad w(a):=\Var _a\Bigl (\frac h\Delta \sum _{m<m_\Delta }\nu _{y_m}\Bigr ), \label {eq:revelation-defect} \end{equation}$
and $w$ is a finite computation from $(P,\nu )$ :
$\begin{equation} w=\Bigl (\frac h\Delta \Bigr )^2\Bigl [\sum _{m<m_\Delta }P^m\Bigl (\nu ^2+2\nu \sum _{d=1}^{m_\Delta -1-m}P^d\nu \Bigr )-\Bigl (\sum _{m<m_\Delta }P^m\nu \Bigr )^2\Bigr ]. \label {eq:revelation-formula} \end{equation}$
It vanishes if and only if the window average is $\cF _k$ -measurable, in particular when $m_\Delta =1$ , and is otherwise strictly positive.

Proof.(a) The normals $(\xi _k)$ are independent of the atom path, so conditionally on $\cG _k$ the return $X_{k+1}-X_k$ is still $N(-\tfrac 12h\nu _{y_k},h\nu _{y_k})$ , whose exponential has conditional mean one; hence $S$ is a $(\cG _k)$ -martingale. The law of $(y,X)$ does not involve the filtration, so marginals and unconditional expectations of payoffs of the pair are unchanged. (b) $A_w-A_u$ is $\cG _0$ -measurable, so its conditional expectation in $(\cG _k)$ is itself; the second identity in (37) is the tower property, $U_{t_k}$ being the conditional expectation of the same variable in the smaller filtration. (c) The defect is then the conditional standard deviation of the window average, which is (38); expanding the square gives (39), since $\E _a[\nu _{y_m}\nu _{y_{m'}}]=\bigl (P^m(\nu \cdot P^{m'-m}\nu )\bigr )(a)$ for $m\le m'$ by the Markov property. A variance vanishes exactly when its argument is degenerate, and for $m_\Delta =1$ the average is $\nu _{y_k}$ itself.∎

Proposition 6.3 (The revelation defect and its refinement limit) . Let $\mathsf r$ be a scheme with the limit of Theorem 5.6, and fix a date $t$ on the common grid. Then $\cR ^{(m)}(t)\to \cR ^\infty (t)$ , the conditional standard deviation of $\Delta ^{-1}(A^\infty _{t+\Delta }-A^\infty _t)$ given $\cG _t$ , which is strictly positive unless that window clock is $\cG _t$ -measurable. Consequently no statement about the primitive law of $(y,X)$ distinguishes a scheme from its revealed twin, while the conditional estimates of Proposition 4.5 and Section 5 do.

Proof.By Theorem 5.6 (b) the clocks converge uniformly and in $L^2$ , so $U^{{\rm rev},(m)}_t\to \Delta ^{-1}(A^\infty _{t+\Delta }-A^\infty _t)$ in $L^2$ . By co-immersion in the filtration of the whole hierarchy (Lemma 4.3) every $U^{(m)}_t$ is the conditional expectation of $U^{{\rm rev},(m)}_t$ in the joint filtration $\cG _t$ , as is the limit’s by Theorem 5.6 (c), so no convergence of filtrations is needed: conditional expectation is an $L^2$ contraction, hence $U^{(m)}_t\to \E [\Delta ^{-1}(A^\infty _{t+\Delta }-A^\infty _t)\mid \cG _t]$ in $L^2$ as well, and with them the defect. The last sentence is Proposition 6.2 (a): primitive path laws and the prices of their bounded measurable payoffs coincide at every resolution, while (38) converges to the stated limit, which is nonzero precisely under the nonmeasurability condition in the proposition.∎

Thus co-immersion in Lemma 4.2 is substantive: it places the two conditional expectations in a common filtration. The defect itself is available from the fitted matrix through (39), and Section 8 computes it along the example.

7Exact panel matching

The rows of Theorem 3.1 are identities of the construction; a declared finite panel of index-option prices is not. This section separates three statements that are easily conflated. At one resolution a feasible capped-call panel can be matched exactly. Across resolutions the resulting objects are controlled-jump perturbations of the original scheme. They are shown to retain the original limit when an additional, checkable decay condition holds for the jumps. An unrelated external panel can therefore be matched level by level, but no convergence claim is made for that fitted family.

Definition 7.1 (Time-inhomogeneous resolution) . A time-inhomogeneous resolution replaces the single matrix $P$ of Definition 2.1 by a sequence $(P_k)_{k<n}$ of stochastic matrices on $\cA$ , the chain moving by $P_k$ at step $k$ . All of Definition 2.3 carries over with $P^{m}$ replaced by the ordered product $P_kP_{k+1}\cdots P_{k+m-1}$ , and Theorem 3.1 holds verbatim, its proof using only the one-step conditional law. The certificates (C1)–(C4) are imposed on every $P_k$ , and the defect (10) on every pair $(P_k,(P'_{2k})(P'_{2k+1}))$ ; the refinement coupling and Lemma 4.4 then hold unchanged, the recursion being step by step.

Definition 7.2 (Controlled-jump perturbation) . Let $\mathsf r$ be a scheme and $J\in \N$ . At level $m$ , on at most $J$ grid steps, after the ordinary transition governed by $P^{(m)}$ , allow an instantaneous correction $C_j^{(m)}$ satisfying

\begin{equation} C_j^{(m)}(a,\{b:|b-a|\le \ell _j^{(m)}\})=1, \qquad \ell ^{(m)}:=\max _{j\le J}\ell _j^{(m)}. \label {eq:controlled-jump} \end{equation}

Thus the kernel of the complete grid step is

P^{(m)}C_j^{(m)}

, and the correction is measured relative to the ordinary step:

W_\rho (P^{(m)}C_j^{(m)}(a,\cdot ),P^{(m)}(a,\cdot ))\le \ell _j^{(m)}

. The resulting time-inhomogeneous family is a controlled-jump perturbation. Its refinement modulus is

\begin{equation} \varepsilon ^{\rm ex}_m:=\varepsilon _m+J\bigl (\ell ^{(m)}+\ell ^{(m+1)}\bigr ). \label {eq:exceptional-modulus} \end{equation}

The bound is obtained by a four-chain star coupling: couple each corrected level to its uncorrected level, and couple the two uncorrected levels by Definition 4.1. Each same-level arm contributes at most the sum of its

J

jumps, while the middle arm contributes

\varepsilon _m

; fresh uniforms for all corrections preserve every marginal and co-immersion in the enlarged joint filtration. Thus no corrected-chain tail is inserted into the middle recursion. The Lyapunov bound is multiplied by at most

\exp \{J\beta (2\ell ^{(m)}\Xi _m+(\ell ^{(m)})^2)\}

. We call the perturbation convergent at order

\rho

when these factors are uniformly bounded and the summability hypothesis of Theorem 5.6 holds with

\varepsilon ^{\rm ex}_m

in place of

\varepsilon _m

. Neither (C1) nor (C4) is imposed on

C_j^{(m)}

or on

P^{(m)}C_j^{(m)}

; (C2) is replaced by the displayed Lyapunov multiplier. The correction displacement is instead charged explicitly in (41). In particular, its contribution to the largest within-cell move is already covered by the weighted summability of

\varepsilon _m^{\rm ex}

Fix grid fixings $T_1<\dots <T_J\le T_{\rm op}$ with $T_1>0$ and $T_{j+1}-T_j\ge \Delta +h$ , put $k_j:=T_j/h$ , and let each fixing carry a finite family of bounded Lipschitz rows $r_{j,1},\dots ,r_{j,K_j}:\R _+\to \R$ with declared values $\tau _{j,\iota }$ . The resolution’s index at a fixing is a function of its atom,

\begin{equation} V_{T_j}=\psi (y_{k_j}),\qquad \psi (a_i):=\Bigl (\frac h\Delta \sum _{m<m_\Delta }(P^m\nu )_i\Bigr )^{1/2}, \label {eq:cir-vix-map} \end{equation}

and the tenor condition $T_{j+1}-T_j\ge \Delta +h$ ensures that the window $[T_j,T_j+\Delta )$ carries no later correction — with the chronology fixed below, the correction for $T_{j+1}$ occupies the step ending at $T_{j+1}$ , which is the reason for the extra $h$ — so that $\psi$ , computed from the uncorrected kernel, is the corrected chain’s index map at every fixing.

Lemma 7.3 (The index map is monotone) . Under (C1) the kernel $P$ is stochastically monotone and $\psi$ is nondecreasing.

Proof.(C1) orders the rows stochastically; a monotone kernel maps nondecreasing vectors to nondecreasing vectors, and $\nu$ is nondecreasing.∎

The correction. On the step ending at $T_j$ the chain first moves by $P$ , reaching the pre-correction atom $y_{k_j-}$ , and then makes one extra transition governed by a stochastic matrix $C^{(j)}$ on $\cA$ , driven by a uniform variable $\vartheta _j$ independent of everything else. The atom the chain holds at $T_j$ is therefore the corrected one, $V_{T_j}$ is read from it, and the chain continues by $P$ . In the notation of Definition 7.1 this is

\begin{equation} P_{k_j-1}:=P\,C^{(j)},\qquad P_k:=P\ \ \text {for }k\notin \{k_j-1\}, \label {eq:correction-placement} \end{equation}

so Theorem 3.1 applies to it. This placement makes the quote depend on the corrected atom at the fixing. On the coupled space of Section 4, enlarge the joint filtration to $\widehat \cG _k:=\cG _k\vee \sigma (\vartheta _j:T_j\le t_k)$ ; both chains remain co-immersed in it, since the $\vartheta _j$ are independent of everything else, so Lemma 4.2 and Proposition 4.5 hold with $\widehat \cG$ in place of $\cG$ . Earlier corrections enter only through $\mu ^{(j)}$ , the law of the pre-correction atom at $k_j$ under the chain already corrected at $T_1,\dots ,T_{j-1}$ , which is computed forward by matrix–vector products.

Put $u_-:=\min _{a\in \cA }\psi (a)^2>0$ . The rows are capped call rows, the case VIX options present: $r_{j,\iota }(V)=(V-K_{j,\iota })^+\wedge \bar M$ with strikes $\sqrt {u_-}<K_{j,1}<\dots <K_{j,K_j}<\bar M$ ; the cap only keeps the rows bounded, as the general setting requires, and may be taken above the chain’s range. Put $K_{j,K_j+1}:=\bar M$ . Let $\varepsilon _{j,\iota }:=\tau _{j,\iota }-\E r_{j,\iota }(\psi (y_{k_j-}))$ be the residuals and $\ell _j\in \delta \N$ a move length. A block for row $\iota$ is a set $I_{j,\iota }\subseteq \{a\in \cA :a+\ell _j\in \cA \}$ of consecutive atoms with $\psi (a),\psi (a+\ell _j)\in (K_{j,\iota },K_{j,\iota +1})$ , the blocks being chosen so that $I_{j,\iota }$ and $I_{j,\iota '}+\ell _j$ are disjoint for $\iota \ne \iota '$ ; when $\psi$ is monotone, as in Lemma 7.3, this is automatic; otherwise disjointness is part of the block certificate. A block and its own shift may overlap. On a block the call rows are linear: $r_{j,\iota '}(\psi (a+\ell _j))-r_{j,\iota '}(\psi (a))$ equals $\psi (a+\ell _j)-\psi (a)$ for $\iota '\le \iota$ and vanishes for $\iota '>\iota$ . Define the two averaged slopes of the VIX map over the block, weighted by $\mu ^{(j)}$ on the block and on its shift,

\begin{equation} s^+_{j,\iota }:=\frac {\sum _{a\in I_{j,\iota }}\mu ^{(j)}(a)\bigl [\psi (a+\ell _j)-\psi (a)\bigr ]}{\ell _j\,\mu ^{(j)}(I_{j,\iota })},\qquad s^-_{j,\iota }:=\frac {\sum _{a\in I_{j,\iota }}\mu ^{(j)}(a+\ell _j)\bigl [\psi (a+\ell _j)-\psi (a)\bigr ]}{\ell _j\,\mu ^{(j)}(I_{j,\iota }+\ell _j)}, \label {eq:cir-slope-matrix} \end{equation}

so that the slope matrix of the up-moves is $\widetilde M^{(j)}_{\iota '\iota }=s^+_{j,\iota }\1\{\iota '\le \iota \}$ , triangular with a bounded inverse. The correction is a two-sided flow with a row-neutral baseline: every atom of a block may move up by $\ell _j$ and every atom of the shifted block may move down by $\ell _j$ , with baseline probabilities chosen so that the two moves change every row by opposite amounts; the residual is absorbed by a signed perturbation of the baseline. Because the columns of the up- and down-slope matrices are proportional for call rows, the baseline cancels exactly at the level of block masses, and no comparison of individual atom masses is needed.

Theorem 7.4 (Exact declared capped call rows at every level) . Let the atom set be a lattice of spacing $\delta$ — as in Lemma 2.7, and as the move lengths $\ell _j\in \delta \N$ already presuppose, so that $a+\ell _j$ is again an atom whenever it remains below the cap; the block definition explicitly requires this. For a general finite $\cA$ the blocks and their shifts have to be defined through the atom indices instead. Suppose $T_{j+1}-T_j\ge \Delta +h$ for all $j$ , and that at each fixing $T_j$ there are $m_j>0$ and $0<s_j\le S_j<\infty$ such that for all $\iota$

\begin{equation} \mu ^{(j)}(I_{j,\iota })\ge m_j,\qquad \mu ^{(j)}(I_{j,\iota }+\ell _j)\ge m_j,\qquad s^\pm _{j,\iota }\in [s_j,S_j], \label {eq:cir-exact-rows-hyp} \end{equation}

with

\begin{equation} \ell _j\ \ge \ \frac {8S_j}{s_j^2m_j^2}\,|\varepsilon _j|_\infty . \label {eq:cir-move-length} \end{equation}

Then there is a correction

C^{(j)}

supported on the blocks and their shifts under which every declared row is exact,

\E r_{j,\iota }(V_{T_j})=\tau _{j,\iota }

for all

\iota

; it moves the atom by at most

\ell _j

. At this fixed level the result is purely algebraic; a statement about a refining corrected family requires the additional decay conditions of Proposition 7.5.

Proof.Fix $j$ and drop the index. Let $T$ be the matrix $T_{\iota '\iota }=\1\{\iota '\le \iota \}$ ; its inverse is bidiagonal with entries $\pm 1$ , so $z:=\widetilde M^{-1}\varepsilon$ satisfies $|z|_\infty \le 2|\varepsilon |_\infty /s$ . Put $\bar p:=sm/(4S)\le \tfrac 14$ , $\bar p^-_\iota :=\bar p\,s^+_\iota \mu (I_\iota )/(s^-_\iota \mu (I_\iota +\ell ))\le \tfrac 14$ by (45), and $q_\iota :=z_\iota /(2\ell \mu (I_\iota ))$ , so that $|q_\iota |\le |\varepsilon |_\infty /(s\ell m)\le \bar p/2$ by (46). Let every atom $a\in I_\iota$ move to $a+\ell$ with probability $\bar p+q_\iota \in [\bar p/2,3\bar p/2]$ , and every atom $a+\ell$ , $a\in I_\iota$ , move to $a$ with probability $\bar p^-_\iota (1-q_\iota /\bar p)\in [\bar p^-_\iota /2,3\bar p^-_\iota /2]$ . All probabilities lie in $(0,\tfrac 38]$ ; an atom that belongs both to a block and to the shift of the same block has two moves, with total probability at most $\tfrac 34$ , and atoms of distinct blocks and shifts are disjoint, so each row of $C$ sums to one and $C$ is stochastic. The change of $\E r_{\iota '}(V_T)$ is the sum over all moves of the moved mass times the change of the row, which on block $\iota$ is ∎

\Bigl [(\bar p+q_\iota )\,\ell \mu (I_\iota )s^+_\iota -\bar p^-_\iota (1-q_\iota /\bar p)\,\ell \mu (I_\iota +\ell )s^-_\iota \Bigr ]\1\{\iota '\le \iota \} =2q_\iota \,\ell \mu (I_\iota )s^+_\iota \1\{\iota '\le \iota \}=z_\iota \widetilde M_{\iota '\iota }

by the definition of

\bar p^-_\iota

and (44), whatever the sign of

z_\iota

; summing over

\iota

gives

(\widetilde Mz)_{\iota '}=\varepsilon _{\iota '}

. The timing condition ensures that the VIX at

T_j

is read from the chain corrected up to

T_j

and moved by no later correction inside its window, so

\psi

is its VIX map and

\mu ^{(j)}

is the relevant law.

Proposition 7.5 (Levelwise matching and preservation of the limit) . The hypotheses (45) of Theorem 7.4 are certificates: the block masses are entries of $\mu ^{(j)}$ , itself obtained by the ordered product through the $k_j$ steps preceding the fixing (including the earlier corrected kernels), and the two averaged slopes are finite sums over a block, so all are computed from $(P,\nu )$ at the resolution in use. Fix $\rho >2$ and let $\mathsf r$ be a scheme of order $4\rho$ satisfying the hypotheses of Theorem 5.6, whose atom sets are $\cA _m=\{\delta _m,2\delta _m,\ldots ,N_m\delta _m\}$ . Declare a panel of capped index call rows at fixings that are grid dates at level $m_0$ (and hence at every finer dyadic level), with $T_{j+1}-T_j\ge \Delta +h_{m_0}$ . Suppose the block certificates hold for all $m\ge m_0$ , with $m_j,s_j,S_j$ independent of $m$ , for the least lattice move

\begin{equation} \ell _j^{(m)}:=\delta _m\left \lceil \frac {8S_j}{s_j^2m_j^2}\frac {|\varepsilon _j^{(m)}|_\infty }{\delta _m} \right \rceil . \label {eq:minimal-panel-move} \end{equation}

When the residual vanishes, take

C^{(j)}=I

and

\ell _j^{(m)}=0

rather than introducing a vacuous block. Then:

(a): at every level $m\ge m_0$ the construction produces a time-inhomogeneous resolution that matches the panel exactly and is a controlled-jump perturbation with jump budget $\ell ^{(m)}:=\max _j\ell _j^{(m)}$ ;
(b): if the declared values are the family’s own limiting values, $\tau _{j,\iota }=\E \,r_{j,\iota }(V^\infty _{T_j})$ with $V^\infty$ the index of Theorem 5.6, put
$d_q^V:=\varepsilon _q^{(2)}+\Lambda _{2n_q},\qquad b_q^V:= \begin {cases} C d_q^V,&\text {if a common index floor is certified},\\ C\sqrt {d_q^V},&\text {otherwise}, \end {cases}$
where $C$ is the uniform moment constant in the fixing-date estimate below, and put $a_m:=\sum _{q\ge m}b_q^V$ . Then, by induction over the finite panel,
$\begin{equation} \max _j\ell _j^{(m)}\le C(a_m+\delta _m). \label {eq:panel-move-upper} \end{equation}$
If the Lyapunov factors in Definition 7.2 are uniformly bounded and its weighted summability condition holds, the corrected family is a convergent controlled-jump perturbation with modulus
$\begin{equation} \varepsilon ^{\rm cor}_m\ \le \ \varepsilon ^{(\rho )}_m+\sum _j\bigl (\ell ^{(m)}_j+\ell ^{(m+1)}_j\bigr ), \label {eq:corrected-modulus} \end{equation}$
and it has the same limit as $\mathsf r$ ;
(c): for a panel declared externally, every corrected resolution still matches exactly, but neither decay of the sequential residuals nor convergence of the corrected family follows. In particular, if one of the sequential residuals stays bounded away from zero, then (47) keeps the corresponding jump bounded away from zero and the same-limit condition fails.

Proof.(a) is Theorem 7.4; (40) follows from the support of $C^{(j)}$ . For (b), a direct fixing-date version of the proof of Theorem 5.3 gives $\E |U^{(q)}_{T_j}-U^{(q+1)}_{T_j}|\le C d_q^V$ : use the $L^2$ refinement recursion, one Cauchy–Schwarz inequality for the squared labels, and the uniform second moments. The common-floor inequality gives the first choice of $b_q^V$ , while $|\sqrt u-\sqrt v|\le \sqrt {|u-v|}$ gives the second. Since $\varepsilon _q^{(2)}\le \varepsilon _q^{(\rho )}$ , the weighted summability in Theorem 5.6 implies $\sum _q b_q^V<\infty$ ; in the floor-free case this follows from Cauchy–Schwarz after writing $\sqrt {d_q^V}=\sqrt {n_q^{1/\rho }d_q^V}\,n_q^{-1/(2\rho )}$ . Thus before the first correction the residual is bounded by the uncorrected resolution-to-limit error $Ca_m$ . Before correction $j$ , the preceding $j-1$ jumps can change a bounded Lipschitz row by at most $C\sum _{i<j}\ell _i^{(m)}$ under the same-resolution coupling. Indeed, (C1) at order $2$ couples $P^q(a,\cdot )$ and $P^q(b,\cdot )$ within $c_m^q|a-b|$ in $L^2$ ; after averaging $q=0,\ldots ,m_\Delta -1$ , the reverse triangle inequality shows $|\psi _m(a)-\psi _m(b)|\le |a-b|$ . The least-lattice choice (47) therefore gives ∎

\ell _j^{(m)}\le C\Bigl (a_m+\sum _{i<j}\ell _i^{(m)}\Bigr )+\delta _m .

A finite induction proves (48). The four-chain coupling of Definition 7.2 gives (49). The proof of Theorem 5.6, rather than the theorem literally, now extends with two substitutions: the star-coupling grid bound

\varepsilon _m^{\rm cor}

replaces the ordinary refinement modulus, and the corrected cell oscillation satisfies

J_{m+1}^{\rm cor}\le J_{m+1}^{\rm ord}+\ell ^{(m+1)}

. The uniform moments come from the displayed Lyapunov multiplier, and the weighted summability in Definition 7.2 covers both new terms. Finally, the two same-level arms of the star coupling compare each corrected resolution with its base resolution; their weighted decay makes the two limits equal. (c) follows because exactness at one level used no inter-resolution hypothesis. Earlier corrections enter later residuals, so no formula for their limit is asserted; a residual bounded away from zero gives the stated obstruction directly from (47).

Remark 7.6 (Two different problems) . Part (b) says that a family can reproduce its own limiting panel exactly at every resolution without changing its limit when the controlled jumps satisfy the same maximal summability demanded of the base refinement. Part (c) says that fitting a panel the family does not already carry is a different problem, and one this paper does not solve. At each fixed resolution the panel is matched exactly, which is what a calibration needs; what is not established is that the corrected family converges, nor what its limit is. Whether a declared panel is carried by some continuous market at all is the feasibility question of Huang [29], and whether a scheme can be constructed to converge to a market carrying it is the natural companion question, left open here.

7.1Fixings closer than a tenor

The tenor condition of Theorem 7.4 keeps every window free of later corrections, so that each fixing is solved on its own, at the law left by the earlier ones. When two fixings are closer than a tenor, the correction of the later one sits inside the window of the earlier one and changes its index map, while the correction of the earlier one changes the law at the later one. The residual map is then a coupled finite-dimensional system, and its solution is a quantitative inverse function theorem whose constants are computed.

Lemma 7.7 (Quantitative inverse) . Let $F:\R ^d\to \R ^m$ be twice continuously differentiable on a neighbourhood of the closed ball $\bar B(0,\rho )$ with $F(0)=0$ , let $J:=DF(0)$ have rank $m$ with least singular value $s>0$ , and let $\|D^2F(c)\|\le L$ for $|c|\le \rho$ , the norm being that of a bilinear map into Euclidean $\R ^m$ . If $|\varepsilon |\le \min \{s^2/(4L),\,s\rho /2\}$ , then there is exactly one $c$ in the ball of radius $2|\varepsilon |/s$ of the row space of $J$ with $F(c)=\varepsilon$ ; it is the limit of the iteration $c_{n+1}=c_n-J^+\bigl (F(c_n)-\varepsilon \bigr )$ from $c_0=0$ , $J^+$ the pseudo-inverse of $J$ . Here and below, a quotient of the form $s^2/(4L)$ is understood as $+\infty$ when $L=0$ .

This is the Newton–Kantorovich theorem [40], in the form the panel solves below require: a rectangular Jacobian read through its pseudo-inverse and its least singular value, a curvature bound on an explicit ball, and the solution located in the row space, so that the three constants entering the threshold are the three a resolution can compute. The proof is given in Appendix 1.

Theorem 7.8 (Joint matching of fixings closer than a tenor) . Let the atom set be a lattice as in Theorem 7.4, let the fixings be grid dates with $T_{j+1}-T_j\ge h$ only, and keep the rows, the blocks $I_{j,\iota }$ , the move lengths $\ell _j\in \delta \N$ and the certificates (45), all evaluated on the uncorrected chain, with block masses $m_{j,\iota }:=\mu ^{(j)}(I_{j,\iota })$ . Declare the flow coefficients from them, $\bar p_j:=s_jm_j/(4S_j)$ and $\bar p^-_{j,\iota }:=\bar p_js^+_{j,\iota }m_{j,\iota }/(s^-_{j,\iota }\mu ^{(j)}(I_{j,\iota }+\ell _j))$ . For $z=(z_{j,\iota })\in \R ^{K}$ , $K:=\sum _jK_j$ , let $C^{(j)}(z_j)$ be the two-sided flow of Theorem 7.4 at fixing $j$ with these coefficients and the signed perturbations $q_{j,\iota }:=z_{j,\iota }/(2\ell _jm_{j,\iota })$ , and let the corrected chain carry $P_{k_j-1}:=P\,C^{(j)}(z_j)$ at every fixing; at $z=0$ every fixing carries its reference two-sided baseline. The index map of the corrected chain at $T_j$ is

\begin{equation} \psi ^z_j(a)^2:=\frac h\Delta \sum _{m<m_\Delta }\bigl (P_{k_j}P_{k_j+1}\cdots P_{k_j+m-1}\nu \bigr )(a), \label {eq:joint-index-map} \end{equation}

which depends on

z_{j'}

exactly for the fixings in

W_j:=\{j':h\le T_{j'}-T_j\le \Delta -h\}

, and the residual map and the residual are

\begin{equation} R_{j,\iota }(z):=\E _z\,r_{j,\iota }\bigl (\psi ^z_j(y_{k_j})\bigr )-\E _0\,r_{j,\iota }\bigl (\psi ^0_j(y_{k_j})\bigr ), \qquad \varepsilon _{j,\iota }:=\tau _{j,\iota }-\E _0\,r_{j,\iota }\bigl (\psi ^0_j(y_{k_j})\bigr ), \label {eq:joint-residual-map} \end{equation}

\E _z

denoting the corrected chain and

\E _0

the chain with the baselines in force. Put

\bar \rho :=\min _{j,\iota }\ell _jm_{j,\iota }\bar p_j

; on the box

|z_{j,\iota }|\le \ell _jm_{j,\iota }\bar p_j

every off-diagonal move probability lies in

(0,\tfrac 38]

and every

C^{(j)}(z_j)

is stochastic. Put

\mathcal K_j:=\{K_{j,\iota },K_{j,\iota }+\bar M:\iota \le K_j\}

, the kink set of the declared capped calls. Suppose:

(i): (strict prescribed index margin) for every $j$ and $a\in \cA$ , the range of $\psi ^z_j(a)$ as $z$ varies over the box is contained in one connected component of $\R _+\setminus \mathcal K_j$ . In addition, for every $a\in I_{j,\iota }$ and every $z$ in the box,
$\begin{equation} \psi ^z_j(a),\ \psi ^z_j(a+\ell _j) \ \in \ (K_{j,\iota },K_{j,\iota +1}). \label {eq:joint-prescribed-cell} \end{equation}$
Thus the baseline anchor $z=0$ and every admissible perturbation retain the open affine cell for which the block was declared. Since $\psi ^z_j(a)^2$ is a polynomial in $z$ , affine in each scalar coordinate, its extreme values over the box are attained at the corners; strict separation of the corner extrema from the boundaries of the prescribed component is a finite check;
(ii): (certificate) $\bar J:=DR(0)$ is invertible with least singular value $\bar s$ , and $\|D^2R(z)\|\le \bar L$ for $|z|\le \bar \rho$ .

If $|\varepsilon |\le \min \{\bar s^2/(4\bar L),\,\bar s\bar \rho /2\}$ , then there is exactly one $z$ with $|z|\le 2|\varepsilon |/\bar s$ under which every declared row is exact, $\E _z\,r_{j,\iota }(V_{T_j})=\tau _{j,\iota }$ for all $j,\iota$ ; it is the limit of the iteration of Lemma 7.7, and each fixing’s correction moves the atom by at most $\ell _j$ . The Jacobian has the block form

\begin{equation} \bar J=\widetilde M+N+U, \label {eq:joint-jacobian} \end{equation}

\widetilde M

block diagonal,

N

strictly block lower triangular and

U

strictly block upper triangular in the order of the fixings:

N_{jj'}

j'<j

, is the dependence of the pre-correction law at

T_j

on the earlier correction;

U_{jj'}

vanishes unless

j'\in W_j

, where it is the dependence of the index map at

T_j

on a later correction inside its window; and the diagonal block has the entries

\begin{equation} \widetilde M^{(j)}_{\iota '\iota }=\frac {\mu ^{(j)}_0(I_{j,\iota })\,s^+_{j,\iota }(0)+(\bar p^-_{j,\iota }/\bar p_j)\,\mu ^{(j)}_0(I_{j,\iota }+\ell _j)\,s^-_{j,\iota }(0)}{2m_{j,\iota }}\,\1\{\iota '\le \iota \}, \label {eq:joint-diagonal} \end{equation}

with

\mu ^{(j)}_0

the pre-correction law at

T_j

and

s^\pm _{j,\iota }(0)

the slopes (44) of

\psi ^0_j

, both under the chain with the baselines in force. When both agree with their uncorrected-chain counterparts, (54) is the slope matrix of Theorem 7.4. The pre-correction law agrees at the first fixing, but its slopes can still be changed by later baselines lying in

W_1

. When

T_{j+1}-T_j\ge \Delta

for all

j

, every

W_j

is empty,

U=0

, and

\bar J

is invertible as soon as its diagonal blocks are; forward substitution is then the fixing-by-fixing triangular mechanism of Theorem 7.4, with the actual pre-correction law at each fixing.

The proof is given in Appendix 2.

Remark 7.9 (Scope of the construction) . The two-sided flow works for any family of rows whose up- and down-slope columns are proportional on each block, which call and put rows are; for general rows the baseline does not cancel, an offset of order $\bar p\,\ell$ enters, and Theorem 7.8 still applies, since it measures the residual against the chain with the baselines in force and uses the cancellation only to identify the diagonal blocks; for rows with finitely many kinks the index margin is taken at the kinks. Fixings closer than a tenor cost the coupling $U$ and the quantitative inverse in place of the triangular solve. Section 11.3 computes $\bar J$ for two fixings half a tenor apart: $U$ is half the size of the diagonal blocks, and the fixing-by-fixing solve leaves on the first fixing a residual of about a sixth of the declared one. In the language of a finite-fibre exactification, the residual fibre has dimension $\sum _jK_j$ , the right inverse is $\bar J^{-1}$ , and the displacement is the move length. What a block flow absorbs at one resolution is bounded: a row moves by at most the mass of a block times the baseline probability times the move length times a slope, so $|\varepsilon |$ is at most of the order $s_jm_j\bar p_j\ell _j$ , and $\ell _j$ is bounded by the strike spacing through the block condition. A residual beyond that is not reached by one block flow at that resolution; along a refining family it is reached once it falls below the bound, which for the family’s own limiting panel is a question of level (Proposition 7.5).

8Certificates and refinement, computed

Kernel certificates, levels, charges, transport defects and index maps below are deterministic finite-matrix computations; coupled path errors and physical probabilities are Monte Carlo diagnostics. The comparison is target-free but not nonparametric: a square-root transition law generates the finite kernels and supplies the closed-form revelation benchmark.

The finite sweep. Use Lemma 2.7 with $\kappa =3$ , $\theta =0.04$ , $\sigma =0.4$ , $y_0=0.2$ , $\Delta =30/365$ , $T_{\rm op}=6\Delta$ and $H=7\Delta$ ; here $d_B=4\kappa \theta /\sigma ^2=3$ . Take $h_m=\Delta /2^m$ , $\delta _m=(0.2/9)/3^m$ and caps approximately $2.4,2.5,2.6,2.7$ , giving $N_m=108,338,1053,3281$ . The nested lattices contain $0.2$ , so the projection terms in (14) are $\iota _{\rho ,m}=\zeta _m=0$ and the coupled chains start at the same atom; moreover $\delta _m\le h_m^{3/2}$ and $\delta _m/h_m^{3/2}\to 0$ .

The reported orders are $\rho =2$ and $8$ . With the computed $C_0=7.65$ , the smallest levels satisfying (13) are

\begin{equation} \ell ^{(2)}_m=0.896, 0.962, 1.023,\qquad \ell ^{(8)}_m=1.700, 1.837, 1.964 . \label {eq:numerics-levels} \end{equation}

With slack $\eta =0.1$ in (17), required cap margins range from $0.19$ to $0.32$ at order $2$ and $0.37$ to $0.65$ at order $8$ , versus available ranges $1.50$ – $1.58$ and $0.64$ – $0.70$ . These levels check the local inputs only. An infinite scheme additionally requires the family conditions and summability of Definition 2.6; a finite table alone cannot certify them.

Exact finite transport. For two atomic laws, the program evaluates $W_\rho ^\rho$ exactly from their masses by north-west-corner transport; upper-tail kernel masses are formed from survival probabilities. The (C4) threshold is obtained by log-sum-exp bisection on $\max _i\sum _jP_{ij}\exp \{\vartheta (a_j-a_i)^2/h\}$ , not by a matrix product.

Table 1. Deterministic neighbouring-level certificates and one Monte Carlo diagnostic. The columns through

\varepsilon _m^{(8)}

are computed from the finite matrices. Here

\vartheta

is the largest row threshold found for that finite resolution, not the analytic constant of Lemma 2.7. The last column is the empirical

L^2

coupled discrepancy from

20\,000

paths.

$m$	$N_m$	$c_{W_2}$	$c_{W_8}$	$(PV)/V$	$\vartheta$	$\ell _m$	$d_{\ell _m}$	$\varepsilon _m$	$\varepsilon ^{(8)}_m$	$\\|y-y'\\|_{L^2}$
0	108	0.9399	0.9846	1.1895	0.68	0.896	$6.1\cdot 10^{-3}$	$1.6\cdot 10^{-1}$	$7.4\cdot 10^{-1}$	$1.5\cdot 10^{-2}$
1	338	0.9695	0.9923	1.0938	1.10	0.962	$2.0\cdot 10^{-3}$	$1.1\cdot 10^{-1}$	$4.7\cdot 10^{-1}$	$7.0\cdot 10^{-3}$
2	1053	0.9846	0.9961	1.0466	1.78	1.023	$6.7\cdot 10^{-4}$	$7.1\cdot 10^{-2}$	$3.1\cdot 10^{-1}$	$3.3\cdot 10^{-3}$

What the certificates show. Stochastic monotonicity holds at every computed level. The contraction factors are in Table 1; the budgets $(1-c_{m+1}^2)/h_{m+1}$ remain in $[1.461,1.494]$ at order $2$ and $[0.374,0.376]$ at order $8$ , and the Lyapunov constants remain controlled. The finite-row (C4) thresholds over all four levels are $0.68,1.10,1.78,2.73$ ; each exceeds the explicit common analytic value $\vartheta _*=3.81\cdot 10^{-4}$ in (18). The numerical thresholds do not strengthen the theorem beyond these levels.

The order- $2$ truncated defects are about $0.27\delta _m$ , below $\delta _m+3\delta _{m+1}$ . The charge $\delta _{m+1}/2$ contributes about $38\%$ of the order- $2$ modulus; omitting it would change $0.163,0.107,0.071$ to $0.101,0.066,0.044$ .

Truncation is material: at order $8$ the all-atom defect rises from $7.7\cdot 10^{-3}$ to $1.3\cdot 10^{-2}$ while its truncated counterpart falls, because overflow lands on different cap atoms. The exact $W_\infty$ defects are $1.64,1.19,0.86$ , and trimming a fixed $10^{-8}$ tail still gives $1.5\cdot 10^{-2},1.3\cdot 10^{-1},2.5\cdot 10^{-1}$ . This supports the averaged metric and explicit high-atom charge used by the theorem.

Coupled diagnostics and revelation. With $20\,000$ paths, the empirical atom discrepancies in Table 1 lie below the certified moduli. Rolling-window readout differences are $6.6\cdot 10^{-3},3.1\cdot 10^{-3},1.5\cdot 10^{-3}$ and the terminal log-spot differences $2.2\cdot 10^{-2},1.5\cdot 10^{-2},1.1\cdot 10^{-2}$ . At $\lambda =1$ , the estimated physical-probability gaps are no larger than their paired standard errors. Sample means of $S_H$ and the density are sanity checks, not evidence for the algebraic martingale identity.

For the family floor in Theorem 5.3, $\min _i\nu _i$ degenerates along the family, $5.5\cdot 10^{-5},6.1\cdot 10^{-6},6.8\cdot 10^{-7}$ , whereas $u_-=\min _iU_i\wedge \min _iU'_i$ rises through $4.9\cdot 10^{-4},2.4\cdot 10^{-3},3.5\cdot 10^{-3}$ . The exact square-root forward averages $(h/\Delta )\sum _{1\le q<m_\Delta }\theta (1-e^{-\kappa qh})$ are $0$ , $2.3\cdot 10^{-3}$ and $3.4\cdot 10^{-3}$ . The comparison concerns four matrices and is not the perturbation bound required for a family certificate.

The revelation defect at the initial atom is $0,7.6\cdot 10^{-3},9.9\cdot 10^{-3},1.10\cdot 10^{-2}$ for windows containing $1,2,4,8$ steps; the one-step zero is exact. They approach the closed-form square-root benchmark $1.2097\cdot 10^{-2}$ while neighbouring index errors fall, separating a persistent information defect from refinement error.

9Boundary test I: leverage

This section is a boundary test, not the paper’s main construction. The base result concerns autonomous finite variance chains; leverage asks which parts of that result survive when volatility also reads the spot. The return rule of Definition 2.2 reads its volatility off the variance atom alone. A local leverage makes it read the spot as well, which a finite object can only do through a proxy. The variance chain stays autonomous, so Lemma 4.4 is untouched; what has to be redone is the spot, where the difference between two resolutions now feeds back into the volatility. The estimate below is a discrete Gronwall inequality on the finer grid, with an exponential weight in the finer chain’s own clock.

Definition 9.1 (Leveraged resolution) . A leveraged resolution adds to Definition 2.1 a proxy grid $\widehat \cX :=\{x_0+i\delta _X:\ |i|\delta _X\le R\}$ with nearest-point quantizer $q_X$ and half-range $R$ (taken to be an integer multiple of $\delta _X$ ), and a leverage $\ell :[0,H]\times \R \to [\ell _-,\ell _+]$ with $0<\ell _-\le \ell _+<\infty$ and

\begin{equation} |\ell (t,x)-\ell (t,x')|\le L_x|x-x'|,\qquad |\ell (t,x)-\ell (t',x)|\le L_t|t-t'| . \label {eq:lsv-leverage} \end{equation}

Its chain carries the state

(y_k,\hat x_k)\in \cA \times \widehat \cX

and moves by

\begin{equation} \sigma _k:=\ell (t_k,\hat x_k)\,y_k,\qquad X_{k+1}-X_k:=-\tfrac 12h\sigma _k^2+\sigma _k\,(W_{t_{k+1}}-W_{t_k}),\qquad \hat x_{k+1}:=q_X\bigl (\hat x_k+X_{k+1}-X_k\bigr ), \label {eq:lsv-chain} \end{equation}

with

\hat x_0=q_X(x_0)

, and is interpolated inside a cell as in Definition 2.2 with

\sigma _k

in place of

\sqrt {\nu _{y_k}}

. Its clock is

A_{t_k}=h\sum _{i<k}\sigma _i^2

and its readouts are those of Definition 2.3 built from that clock, the budget field being a conditional expectation over the pair.

Theorem 9.2 (Exact structure with leverage) . Under $\Q$ , $S=e^X$ is a strictly positive martingale, the log-contract identity holds with $\sigma _k^2$ in place of $\nu _{y_k}$ , and the budget tower is exact. Define the leveraged density and physical law by

L:=\prod _{k<n}\exp \!\left (\lambda \sigma _k (W_{t_{k+1}}-W_{t_k})-\tfrac 12\lambda ^2h\sigma _k^2\right ), \qquad \Pp :=L\Q .

Then

\E L=1

, its density process is the corresponding partial product, the autonomous

y

-chain has the same law under

\Pp

and

\Q

, and

X_{k+1}-X_k\mid \cF _k \ \sim \ \ N\bigl ((\lambda -\tfrac 12)h\sigma _k^2,h\sigma _k^2\bigr ) \quad \hbox {under }\Pp .

Consequently every Bayes row is an exact finite sum of cell-restricted Gaussian expectations. The joint

(y,\hat x)

law need not be unchanged. With genuinely state-dependent leverage, the marginals of

X

also need not be Gaussian mixtures given the atom path: conditioning on the proxy path restricts each increment to a cell. Thus finite-Gaussian-mixture closure is not asserted here; only conditional Gaussianity given the current state is used. Constant or time-only leverage is a special case in which Gaussian-mixture closure can remain.

Proof.Given $\cF _k$ , the return is $N(-\tfrac 12h\sigma _k^2,h\sigma _k^2)$ because $\sigma _k$ is $\cF _k$ -measurable. Conditional exponential normalization proves martingality of $S$ ; conditional expectation of the drift proves the log-contract identity; and the budget statements are the corresponding conditional-expectation identities. At a fixed resolution $\sigma$ is bounded and predictable, so the displayed stochastic exponential is a true martingale and its conditional expectations are its partial products. Conditionally on the complete autonomous $y$ -path, the same bounded stochastic exponential still has mean one; hence changing measure leaves the $y$ -path law unchanged. Completing the square in the next Gaussian increment gives the stated physical mean shift. The Bayes identity then follows by conditioning over the finite state paths and the associated cell-restricted Gaussian integrals.∎

9.1The proxy error

Lemma 9.3 (Proxy error and range) . Let the autonomous variance component of the leveraged resolution satisfy (C2)–(C3), and let $\pi _k:=|\hat x_k-X_k|$ and let $E^R:=\bigl \{\max _{k\le n}|X_k-x_0|\le R-H\delta _X/(2h)\bigr \}$ , where $R>H\delta _X/(2h)$ . On $E^R$ no clamping occurs and $\pi _k\le k\delta _X/2\le H\delta _X/(2h)$ for all $k\le n$ , surely. Moreover, for every $\rho \ge 1$ ,

\begin{equation} \bigl \|\max _{k\le n}\pi _k\bigr \|_{L^\rho } \le \frac {H\delta _X}{2h}+\Bigl (R+\bigl \|\max _k|X_k-x_0|\bigr \|_{L^{2\rho }}\Bigr )\,\Q (E^R{}^c)^{1/(2\rho )}, \label {eq:lsv-proxy} \end{equation}

and

\|\max _k|X_k-x_0|\|_{L^\rho }\le \tfrac 12\ell _+^2H\|\max _k\nu _{y_k}\|_{L^\rho }+C_\rho \ell _+\sqrt H\,\|\max _k\nu _{y_k}\|^{1/2}_{L^\rho }

is bounded by Lemma 5.1. For a family with the uniform constants of (S2)–(S3), write

m_{2\rho }

for the resulting uniform bound at order

2\rho

. The complement of

E^R

has a sub-exponential tail with explicit constants: writing

\beta ':=\beta /(\ell _+^2H)

\begin{equation} \Q \Bigl (\max _{k\le n}|X_k-x_0|>R\Bigr )\ \le \ (2C_0+2)\,e^{-c_RR}, \qquad c_R:=\frac {2\beta '\sqrt {\beta '/2}}{2\beta '+\sqrt {\beta '/2}}, \label {eq:lsv-tail} \end{equation}

so that

\Q (E^R{}^c)\le (2C_0+2)\exp \{-c_R(R-H\delta _X/(2h))\}

. This is the uniform sub-exponential estimate supplied by the stated Lyapunov information; those hypotheses alone do not supply a uniform sub-Gaussian estimate, since a Gaussian martingale evaluated on an exponentially-tailed clock can itself have exponential tails. At any one finite resolution the atom cap also gives a resolution-dependent sub-Gaussian bound, but that bound is not uniform when the caps grow. At order

\rho

the range rule is

\begin{equation} \bigl (R_m+m_{2\rho }\bigr )\,\Q (E^{R_m}{}^c)^{1/(2\rho )} \ \le \ \frac {H\delta _X}{2h_m}, \label {eq:lsv-range-rule} \end{equation}

under which the second term of (58) is no larger than the first. By (59) the rule holds as soon as

\begin{equation} R_m-\frac {H\delta _X}{2h_m}\ \ge \ \frac {2\rho }{c_R} \log \Bigl ((2C_0+2)^{1/(2\rho )}\bigl (R_m+m_{2\rho }\bigr )\frac {2h_m}{H\delta _X}\Bigr ), \label {eq:lsv-range-sufficient} \end{equation}

and under

\delta _X\asymp h_m^{3/2}

the right-hand side is

(\rho /c_R)\log (1/h_m)+O(\log \log (1/h_m))

: for every

\epsilon >0

the range

R_m:=(1+\epsilon )(\rho /c_R)\log (1/h_m)

satisfies the rule for all large

m

. The range must grow, and it grows logarithmically in the mesh, at a rate set by the tail constant

c_R

. The constants in (59) are those of Lemma 5.1 and of Doob’s inequality and are far from sharp; any proved bound

q_m(R)\ge \Q (E^R{}^c)

may replace (59) in (60), and Section 11.2 reports what the displayed constants ask for.

The proof is given in Appendix 3.

9.2The spot between two resolutions

Couple two leveraged resolutions as in Definition 4.1: the variance chains through the quantile level of the finer two-step move, the returns through the shared Brownian motion. Since the variance kernel does not see the spot, Lemma 4.4 applies verbatim and $\|e_k\|_{L^\rho }\le \varepsilon ^{(\rho )}_m$ for every $k$ , where $e_k:=|y_k-y'_{2k}|$ .

Work on the finer grid, and let $D_j:=X_{t'_j}-X'_{t'_j}$ be the difference of the interpolated log-spots at the fine dates, $\bar \sigma _j:=\sigma _{k(j)}$ the coarse volatility in force there, $k(j):=\lfloor j/2\rfloor$ . Both chains start at $x_0$ , so $D_0=0$ . From (57) and the interpolation,

\begin{equation} D_{j+1}-D_j=\Delta _j+b_j\,\Delta W'_j, \qquad \Delta _j:=-\tfrac 12h'\bigl (\bar \sigma _j^2-(\sigma '_j)^2\bigr ), \quad b_j:=\bar \sigma _j-\sigma '_j, \label {eq:lsv-D-recursion} \end{equation}

with $\Delta W'_j:=W_{t'_{j+1}}-W_{t'_j}$ , so the whole estimate rests on the volatility discrepancy $b_j$ . The coarse leverage is read at the coarse date and at the coarse proxy, which tracks the coarse log-spot at $t_{k(j)}$ while $D_j$ compares the chains at $t'_j$ ; writing $o_k:=\sup _{t\in [t_k,t_{k+1}]}|X_t-X_{t_k}|$ for the coarse chain’s cell oscillation, (56) therefore splits $b_j$ into a part proportional to $D_j$ and a part that does not involve it:

\begin{equation} \bigl |b_j\bigr |\ \le \ L_x\,y'_j\,|D_j|\ +\ g_j, \qquad g_j:=\ell _+\bigl (e_{k(j)}+|y'_{2k(j)}-y'_j|\bigr ) +L_xy'_j\bigl (\pi _{k(j)}+o_{k(j)}+\pi '_j\bigr )+L_thy'_j . \label {eq:lsv-g} \end{equation}

Lemma 9.4 (Spot refinement under leverage) . Let $q\ge 2$ be a moment order, let $r,r'$ be neighbouring leveraged resolutions whose autonomous variance components are $r_m,r_{m+1}$ of a scheme of order $q$ , and suppose they use the same leverage function with the common bounds and Lipschitz constants of (56). Put

\begin{equation} \Gamma _j:=\kappa _q\sum _{i<j}h'\bigl ((y'_i)^2+y_{k(i)}^2\bigr ), \qquad \kappa _q:=c_q(1+L_x+\ell _+)^q, \label {eq:lsv-weight} \end{equation}

\begin{equation} E_K:=\Bigl \{h'\!\!\sum _{j<2n}\!\bigl ((y'_j)^2+y_{k(j)}^2\bigr )\le K,\quad \max _{j<2n}\nu _{y'_j}\vee \max _{k<n}\nu _{y_k}\le 1/h'\Bigr \}, \label {eq:lsv-event} \end{equation}

with

c_q

depending only on

q

. Then

\begin{equation} \max _{j\le 2n}\bigl \|D_j\,\1_{E_K}\bigr \|_{L^q} \ \le \ C_{q,H}\,e^{\kappa _q K/q}\,H^{1/q}\,\max _{j<2n}\bigl \|g_j\bigr \|_{L^q}, \label {eq:lsv-spot-error} \end{equation}

and

\Q (E_K^c)\le C_0\bigl (2e^{-\beta K/(2H)}+2e^{-\beta /h'}\bigr )

by Lemma 5.1. Under the budgets

\delta _X\le h^{3/2}

at both meshes and (60) at order

2q

\begin{equation} \max _{j<2n}\bigl \|g_j\bigr \|_{L^q}\ \le \ C\bigl (\varepsilon ^{(q)}_m+\Lambda _{2n}+h^{1/2}\bigr ), \label {eq:lsv-g-bound} \end{equation}

so the right-hand side of (66) is

O\bigl (e^{\kappa _qK/q}(\varepsilon ^{(q)}_m+\Lambda _{2n})\bigr )

The proof is given in Appendix 4.

9.3Refinement of the readouts

Theorem 9.5 (Refinement with leverage) . Let $\rho >1$ and let $r,r'$ be neighbouring leveraged resolutions whose autonomous variance components are $r_m,r_{m+1}$ of a scheme of order $2\rho$ . Suppose they use the same leverage function with the common bounds and Lipschitz constants of (56), their proxy grids satisfy $\delta _X\le h^{3/2}$ at both meshes, and their ranges obey the rule (60) at order $4\rho$ . For every grid-aligned window $(u,w)$ and every $K>0$ ,

\begin{equation} \Bigl \|\max _{k:\,t_k\le u}\bigl |B(t_k;u,w)-B'(t_k;u,w)\bigr |\Bigr \|_{L^\rho } \le d_\rho \,(w-u)\Bigl (C_K\bigl (\varepsilon ^{(2\rho )}_m+\Lambda _{2n}\bigr ) +M_{2\rho }\,\Q (E_K^c)^{1/(2\rho )}\Bigr ), \label {eq:lsv-budget-rate} \end{equation}

where

C_K:=C\,e^{\kappa _{2\rho }K/(2\rho )}

and

M_{2\rho }:=\|\max _k\sigma _k^2\|_{L^{4\rho }}+\|\max _j(\sigma '_j)^2\|_{L^{4\rho }}

. For this fixed window, the index itself follows with the floor

\ell _-^2u_-

as in (28). The terminal log-spot bound of Lemma 9.4 is localized on

E_K

For a quantitative fixing-date physical comparison, assume in addition that $\rho \ge 4$ and that the proxy ranges obey (60) at order $16$ . The order- $2\rho$ scheme then supplies the order- $8$ coupling estimate. With $b_j:=\bar \sigma _j-\sigma '_j$ and

\begin{equation} \mathfrak r_{m,K}:=e^{\kappa _8K/8} \bigl (\varepsilon _m^{(8)}+\Lambda _{2n}\bigr ) +\Q (E_K^c)^{1/8}, \label {eq:lsv-physical-rate} \end{equation}

the localization-and-tail split gives

\begin{equation} \max _{j<2n}\|b_j\|_{L^4(\Q )}+\|D_{2n}\|_{L^2(\Q )} \le C_H\mathfrak r_{m,K}. \label {eq:lsv-b-L4} \end{equation}

Let

L,L'

be the two leveraged densities and let

\Pp =L\Q

\Pp '=L'\Q

. If

8\lambda ^2\ell _+^2H<\beta

, put

\mathcal L_2:=\max \{\|L\|_{L^2(\Q )},\|L'\|_{L^2(\Q )}\}<\infty

. Then

\begin{equation} \begin {aligned} H(\Pp \,|\,\Pp ')&\le C_H\lambda ^2\mathcal L_2H\mathfrak r_{m,K}^2,\\ \|\Pp -\Pp '\|_{\rm TV}&\le C_H|\lambda |\mathcal L_2^{1/2}\sqrt H\,\mathfrak r_{m,K}, \end {aligned} \label {eq:lsv-physical-tv} \end{equation}

and, for every grid date and every

f

bounded by

M

and

L_f

-Lipschitz,

\begin{equation} \begin {split} \E _{\Pp }\bigl |\E _{\Pp }[f(X_H)\mid \cF _k] -\E _{\Pp '}[f(X'_H)\mid \cF '_{2k}]\bigr | \le C_H\bigl (L_f\mathcal L_2 +M|\lambda |\mathcal L_2^{1/2}\sqrt H\bigr )\mathfrak r_{m,K}. \end {split} \label {eq:lsv-physical-readout} \end{equation}

The extra moment order is genuine: the leveraged volatility reads the proxy, hence the return noise, so the atom-path reduction of Proposition 5.5 is unavailable. The density is also a genuine stochastic exponential rather than a conditionally Gaussian product, so

\E L^2=\E e^{\lambda ^2A_H}

is false. What holds is

\begin{equation} \E L^2\ \le \ \bigl (\E \,\mathcal E(4\lambda {\textstyle \int }\sigma \,dW)_H\bigr )^{1/2} \bigl (\E e^{6\lambda ^2A_H}\bigr )^{1/2}\ \le \ \bigl (\E e^{6\lambda ^2A_H}\bigr )^{1/2}, \label {eq:lsv-density-moment} \end{equation}

the first factor being

1

once

\mathcal E(4\lambda \int \sigma \,dW)

is a martingale, for which Novikov asks

\E e^{8\lambda ^2A_H}<\infty

; since

A_H\le \ell _+^2H\max _k\nu _{y_k}

, Lemma 5.1 gives both when

8\lambda ^2\ell _+^2H<\beta

. This is the one place where leverage costs a condition on

\lambda

. No physical rate is claimed under the basic

\rho >1

hypotheses alone. Neither this fixed-window estimate nor the localized spot estimate is a uniform-in-time rolling-index theorem. Such a result would require a common horizon localization and a separate maximal argument, which is not claimed here.

The proof is given in Appendix 5.

Remark 9.6 (What leverage costs, and what it does not) . It costs a second finite coordinate whose mesh must be finer than the step and whose range must grow, logarithmically in the mesh, an exponential weight in the two chains’ clocks, and hence a localization on those clocks, with a tail that Lemma 5.1 bounds. The exponent involves the leverage’s Lipschitz constant — quadratically at $q=2$ , polynomially in general — and not its curvature: the estimate is a discrete Gronwall on the finer grid, so no second derivative of $\ell$ is needed and $\ell$ may be merely Lipschitz. The state bound $\nu \le 1/h'$ inside $E_K$ is used once, to absorb the $q$ -th order term of the one-step expansion, and costs only $e^{-\beta /h'}$ in probability. Leverage does not change the autonomous variance-chain estimate and does not cost the exact rows at any fixed resolution. A quantitative physical comparison additionally costs the order- $8$ coupling and order- $16$ proxy range used in (70). The results above are pairwise, fixed-window and localized where stated; they do not promote the leveraged resolutions to an infinite scheme or assert the full limit theorem of Section 5.

10Boundary test II: marked correlation and joint panels

Joint SPX/VIX calibrations commonly require dependence between SPX returns and the variance state from which the VIX budget is computed. Thus the observable problem is SPX/VIX, while the primitive mechanism introduced here is return–variance dependence. That dependence is not inherited from a continuous model but supplied as further data in a resolution: a mark carried by the return and read off the move of the atom. The variance chain is untouched, so its refinement bound stands; what changes is the return rule, the node correction that keeps the rows exact, and the certificates, which must now cover the mark.

Definition 10.1 (Marked resolution) . A marked resolution adds to Definition 2.1 a mark matrix $\Xi =(\Xi _{ij})$ and a return variance $\alpha =(\alpha _i)$ , $\alpha _i>0$ . Write

\begin{equation} \bar \Xi _i:=\sum _jP_{ij}\Xi _{ij},\qquad K_i:=\log \sum _jP_{ij}e^{\Xi _{ij}},\qquad \mu _{ij}:=\Xi _{ij}-K_i-\tfrac 12\alpha _i,\qquad \Delta A_i:=\alpha _i+2\bigl (K_i-\bar \Xi _i\bigr ), \label {eq:corr-node} \end{equation}

and let the chain move its atom by

P

and its log-spot, given the move from

i

j

, by

\begin{equation} X_{k+1}-X_k:=\mu _{y_ky_{k+1}}+\sqrt {\alpha _{y_k}/h}\,\bigl (W_{t_{k+1}}-W_{t_k}\bigr ). \label {eq:corr-return} \end{equation}

Its clock is

A_{t_k}:=\sum _{i<k}\Delta A_{y_i}

and its readouts are those of Definition 2.3 built from that clock. Write

\begin{equation} \mathsf a_i:=\Delta A_i/h,\qquad \mathsf s_i:=\alpha _i/h \label {eq:corr-rates} \end{equation}

for the clock rate and the return-variance rate, both vectors indexed by atoms.

The three definitions in (74) are the node correction: $K$ normalizes the martingale row, the shift by $-\tfrac 12\alpha$ completes it, and the clock is then read off the log-contract row rather than posited. Jensen’s inequality gives $K_i\ge \bar \Xi _i$ , so $\Delta A_i\ge \alpha _i$ and $\mathsf a\ge \mathsf s$ .

Theorem 10.2 (Exact common-budget structure with correlation) . For a marked resolution, and for every $P$ , $\Xi$ and $\alpha$ ,

\begin{equation} \E \bigl [e^{X_{k+1}-X_k}\mid \cF _k\bigr ]=1, \qquad \E \bigl [X_{k+1}-X_k\mid \cF _k\bigr ]=-\tfrac 12\Delta A_{y_k}, \label {eq:corr-two-rows} \end{equation}

S=e^X

is a positive martingale, the diagonal log-contract identity holds exactly with the clock of Definition 10.1, and the budget tower is exact. Given the whole atom path,

X_k

is Gaussian with mean

\sum _{i<k}\mu _{y_iy_{i+1}}

and variance

\sum _{i<k}\alpha _{y_i}

, so every marginal is a finite Gaussian mixture. Its conditional mean and variance are not

(-\tfrac 12A_{t_k},A_{t_k})

, so the density and Bayes rows of Theorem 3.1(d) are not inherited: their product form used a return whose variance is the clock increment. A physical law for a marked resolution must be specified separately, and its Bayes rows then follow only for that separately specified density.

Proof.Given $\cF _k$ and the next atom $j$ the return is $N(\mu _{ij},\alpha _i)$ , so $\E [e^{\Delta X}\mid \cF _k]=e^{-K_i}\sum _jP_{ij}e^{\Xi _{ij}}=1$ and $\E [\Delta X\mid \cF _k]=\bar \Xi _i-K_i-\tfrac 12\alpha _i=-\tfrac 12\Delta A_i$ . The remaining rows are those of Theorem 3.1, whose proofs used only (77); given the atom path the returns are independent Gaussians, which is the mixture statement.∎

Remark 10.3 (The clock is the budget, not the bracket) . Let $N$ be the martingale part of $X$ under $\Q$ . From (74),

\begin{equation} \Delta \langle N\rangle _i=\alpha _i+\Var _i(\Xi ), \qquad \Delta A_i=\alpha _i+2\bigl (\log \E _ie^{\Xi }-\E _i\Xi \bigr ), \label {eq:corr-bracket-gap} \end{equation}

with

\E _i,\Var _i

over the next atom from

i

: writing

\kappa _p(i)

for the cumulants of

\Xi

under

P(i,\cdot )

, the exact gap is

\begin{equation} \Delta A_i-\Delta \langle N\rangle _i =2\bigl (\log \E _i e^\Xi -\E _i\Xi \bigr )-\Var _i(\Xi ). \label {eq:corr-exact-gap} \end{equation}

If the Taylor series of

t\mapsto \log \E _i e^{t\Xi }

about zero converges at

t=1

, this gap is

\sum _{p\ge 3}\tfrac 2{p!}\kappa _p(i)

; without that analytic condition the series is only the formal small-mark expansion. Thus the two agree to second order in the mark, while the exact discrepancy is (79). This is the finite counterpart of the separation, familiar from the log-contract replication of Neuberger [38] and Demeterfi et al. [23], between the variance a log contract prices and the realized quadratic variation, whose leading correction is likewise a third-cumulant term [15]; a marked resolution carries that separation exactly, and at a size its matrices report. The structural rows of Huang [29] are the three conditional-mean rows, with the clock defined by the log-contract row, so a marked resolution satisfies them exactly; the structural pricing laws of Huang [30] require in addition that

N^2-A

be a martingale, which by (79) holds only if that exact gap vanishes at every atom — under the displayed convergence condition, equivalently if the higher-cumulant combination cancels — for the mark of Remark 10.7 they do not: on the family of Section 11.1 the two clock rates differ by up to six, four and two parts in a thousand at the three coarse resolutions, a gap that shrinks with the mesh but vanishes at none — and redefining

A:=\langle N\rangle

would restore that row while destroying the log-contract row. A marked scheme is therefore guaranteed to lie in the common-budget class; it lies in the linked-law class only under the additional atomwise zero-gap condition.

10.1Marks that do not accumulate

Two resolutions are coupled as in Definition 4.1. Summing (75) shows what the spot difference is made of, and it is not a sum of per-step mark discrepancies: those would accumulate over $H/h$ steps and destroy any rate. What is needed is that the marks telescope, which is a structural condition on the data.

Definition 10.4 (Gradient form and marked-scheme certificates) . A mark is in gradient form if there are vectors $\varphi ,\psi$ with

\begin{equation} \Xi _{ij}=\varphi _j-\varphi _i+\psi _i\qquad \text {for all }i,j, \label {eq:gradient-form} \end{equation}

in which case, with

\mathsf m_i:=(\psi _i-K_i-\tfrac 12\alpha _i)/h

\begin{equation} X_{t_k}=x_0+\varphi _{y_k}-\varphi _{y_0}+h\sum _{i<k}\mathsf m_{y_i} +\sum _{i<k}\sqrt {\mathsf s_{y_i}}\,\bigl (W_{t_{i+1}}-W_{t_i}\bigr ). \label {eq:corr-spot-decomposition} \end{equation}

A marked scheme is a scheme in the sense of Definition 2.6 whose resolutions carry marks in gradient form. For each neighbouring pair and each

\mathsf v\in \{\varphi ,\mathsf m,\mathsf a,\sqrt {\mathsf s}\}

, compute

\begin{equation} \begin {aligned} L_m^{\mathsf v}&:=\max _{\substack {a\ne b\\a,b\le \ell _m+\zeta _m}} \frac {|\mathsf v(a)-\mathsf v(b)|}{|a-b|} \vee \max _{\substack {a'\ne b'\\a',b'\le \ell _m+\zeta _m}} \frac {|\mathsf v'(a')-\mathsf v'(b')|}{|a'-b'|},\\ d_m^{\mathsf v}&:=\max _{a\le \ell _m} \bigl |\mathsf v(a)-\mathsf v'(\pi '(a))\bigr |,\\ R_m^{\mathsf v}&:=\max _a|\mathsf v(a)|\vee \max _{a'}|\mathsf v'(a')| . \end {aligned} \label {eq:mark-certificates} \end{equation}

Empty maxima in the definition of

L_m^{\mathsf v}

are set to zero. The level must pay for excursions above it through the mark charge

\begin{equation} \chi ^{\mathsf v}_m:=R_m^{\mathsf v}\bigl (2C_0e^{-\beta \ell _m^2}\bigr )^{1/\rho } \qquad \text {is required to satisfy}\qquad \chi ^{\mathsf v}_m\le d_m^{\mathsf v}+\tfrac 12\delta _{m+1}. \label {eq:mark-charge} \end{equation}

The inequality is the requirement. If the four ranges have at most polynomial growth in the atom cap, a level of logarithmic size can be enough, but that growth is not automatic. Finally, writing

\begin{equation} \eta _m^{(\rho )}:=\varepsilon _m^{(\rho )}+\Lambda _{2n} +\sum _{\mathsf v}\left [d_m^{\mathsf v}+\chi _m^{\mathsf v} +L_m^{\mathsf v}\bigl (\varepsilon _m^{(\rho )}+\Lambda _{2n}\bigr )\right ], \qquad \sum _m\eta _m^{(\rho )}<\infty , \label {eq:marked-modulus} \end{equation}

is required at the order used below. This weighted summability, rather than decay of a few displayed defects, is what promotes marked resolutions to an infinite scheme. Restricting the Lipschitz constants to the level is not a convenience: for the mark of Remark 10.7 the global constants of

\mathsf a

and

\mathsf m

carry cap effects that the localized comparison need not pay.

Lemma 10.5 (The accumulated mark difference) . Let $r,r'$ be neighbouring resolutions of a marked scheme, coupled as above, and let $\rho \ge 2$ . Write $E_\ell :=\{\max _{k\le n}y_k\vee \max _{j\le 2n}y'_j\le \ell _m\}$ for the event that neither chain leaves the level at which (82) is computed, so that $\Q (E_\ell ^c)\le 2C_0e^{-\beta \ell _m^2}$ by Lemma 5.1. If $J':=\max _{j<2n}|y'_{j+1}-y'_j|$ , then

\begin{align} \bigl \|X_{t_n}-X'_{t_n}\bigr \|_{L^\rho } &\le C_\rho \Bigl [ d_m^\varphi +\chi _m^\varphi +L_m^\varphi \bigl (\varepsilon _m^{(\rho )}+\zeta _m\bigr ) \\ &\quad +H\{d_m^{\mathsf m}+\chi _m^{\mathsf m} +L_m^{\mathsf m}(\varepsilon _m^{(\rho )}+\Lambda _{2n})\} \\ &\quad +\sqrt H\{d_m^{\sqrt {\mathsf s}}+\chi _m^{\sqrt {\mathsf s}} +L_m^{\sqrt {\mathsf s}}(\varepsilon _m^{(\rho )}+\Lambda _{2n})\}\Bigr ] \le C_{\rho ,H}\eta _m^{(\rho )}. \label {eq:corr-spot-Lrho} \end{align}

The complement of $E_\ell$ is included through the charges; the conclusion is an unconditional $L^\rho$ estimate, not a pathwise estimate with an $L^\rho$ error substituted into it.

The proof is given in Appendix 6.

Without (80) the first two groups are replaced by a sum of $H/h$ per-step discrepancies, and no cancellation is available; gradient form is exactly the structure that makes the mark a difference of a potential plus a rate, so that only the endpoints and a Riemann sum survive.

10.2Refinement of the readouts

Theorem 10.6 (Refinement with correlation) . Let $r,r'$ be neighbouring resolutions of a marked scheme at order $\rho \ge 2$ . Then:

(i): for every grid-aligned window,
$\begin{equation} \Bigl \|\max _{k:\,t_k\le u}\bigl |B(t_k;u,w)-B'(t_k;u,w)\bigr |\Bigr \|_{L^\rho } \le d_\rho (w-u)\Bigl [d_m^{\mathsf a}+\chi _m^{\mathsf a} +L_m^{\mathsf a}\bigl (\varepsilon ^{(\rho )}_m+C\Lambda _{2n}\bigr )\Bigr ], \label {eq:corr-budget-rate} \end{equation}$
and the corresponding fixed-window index obeys the square-root estimate when $\min _i\mathsf a_i>0$ . This is an unconditional but pairwise fixed-window bound; no uniform rolling-index statement is inferred from it;
(ii): the terminal log-spot and its Lipschitz prices refine as in Lemma 10.5. If the physical law is separately defined by the stochastic-exponential density with integrand $\lambda \sqrt {\mathsf s}$ , its fixing-date predictions obey the analogue of Proposition 5.5, with integrand difference $\sqrt {\mathsf s_{y_{k(\cdot )}}}-\sqrt {\mathsf s'_{y'_\cdot }}$ . No restriction on $\lambda$ is needed for this finite-pair comparison; this choice does not restore the density rows excluded by Theorem 10.2;
(iii): let the atom set be a lattice and let the finite index map $\psi _n:=(\Delta ^{-1}h\sum _{m<m_\Delta }P^m\mathsf a)^{1/2}$ be nondecreasing in the atom. For fixings separated by at least $\Delta +h$ , suppose the grid dates, capped rows, strikes and cap obey the setup of Theorem 7.4 and, for each sequential pre-correction law $\mu ^{(j)}$ left by the earlier corrections, suppose there are admissible lattice move lengths and disjoint shifted blocks satisfying the block-mass and positive up/down averaged-slope bounds (45) and the residual capacity condition (46). Then Theorem 7.4 applies with $\psi _n$ in place of the map built from $\nu$ . For fixings closer than a tenor, suppose instead all the hypotheses of Theorem 7.8, including its lattice and grid-date setup, admissible blocks and flows, index-margin and Jacobian certificates, and residual threshold. Then that theorem applies with $\mathsf a$ in place of $\nu$ . In either case the declared panel of capped index call rows is matched exactly. A correction is an extra transition after the $P$ -move of its step: the return of the step is generated by the $P$ -move and read, through the mark, at the pre-correction atom, and the extra transition carries no return. Every node quantity of (74), and with it every rate in (76), is therefore that of the uncorrected resolution at every step; the correction changes the atom path and nothing else. The corrected object is a marked time-inhomogeneous resolution on the enlarged path that records the pre-correction atom. The one-step identities of Theorem 10.2 hold conditional on the pre-step filtration, after averaging that recorded atom under the $P$ -row. The corrected marked family is not claimed to be a marked scheme, and no refinement, spot-transfer or physical-transfer statement is asserted for it.

The proof is given in Appendix 7.

Remark 10.7 (The square-root mark) . For the scheme of Lemma 2.7 and a parameter $\varrho \in (-1,1)$ , take

\begin{equation} \varphi _i:=\frac \varrho \sigma \nu _i,\qquad \psi _i:=\frac \varrho \sigma \bigl (\kappa h\nu _i-\kappa \theta h\bigr ),\qquad \alpha _i:=(1-\varrho ^2)h\nu _i , \label {eq:corr-sqrt-mark} \end{equation}

so that

\Xi _{ij}=\varphi _j-\varphi _i+\psi _i=(\varrho /\sigma )(\nu _j-\nu _i-\kappa \theta h+\kappa h\nu _i)

is a square-root-inspired, Euler-correlated mark in gradient form. Its potential

\varphi =(\varrho /\sigma )\nu

is the shift that removes the correlation in the square-root model, the coordinate on which the exact and quadratic-exponential simulation schemes of Broadie and Kaya [16] and Andersen [3] are built; what (80) isolates is the property of that shift the estimates here use, that the mark is a difference of a potential plus a rate, so only endpoints and a Riemann sum survive the sum along the path. On a bounded level,

L_m^\varphi \le 2|\varrho |(\ell _m+\delta _m)/\sigma

and

L_m^{\sqrt {\mathsf s}}=\sqrt {1-\varrho ^2}

; the constants for

\mathsf m

and

\mathsf a

are computed from the normalized rows. The defects across neighbouring resolutions are of two kinds. Those of

\varphi

and

\sqrt {\mathsf s}

are controlled by the projection spacing and vanish exactly on the nested atoms used in Section 11.1. Those of

\mathsf m

and

\mathsf a

carry the row normalization

K

, a logarithm of a sum against the row rather than a function of the atom alone, so they are controlled by the kernel defect and not by the spacing, and are measured to decay more slowly in Section 11.1. Monotonicity of the resulting index map, when needed for the panel statement, is a separate finite check. Nothing in Theorem 10.6 otherwise uses this form: a mark is admissible data, and its certificates are computed.

10.3Spot rows and the joint panel

A declared panel of capped spot call rows is matched by a correction of a different kind. The index rows read the atom chain and the clock; the spot rows read, in addition, the return data. A correction of the marks on one step that leaves the clock increment of every atom unchanged moves spot rows and no index row.

Definition 10.8 (Clock-preserving mark chart) . Let $r$ be a marked resolution, $T>0$ a grid date and $I\subseteq \cA$ a declared set of atoms. A clock-preserving mark chart on the step ending at $T$ is a smooth map $c\mapsto (\Xi (c),\alpha (c))$ on a neighbourhood of a certified closed box $\{|c_\kappa |\le \rho _c,\ \kappa \le d\}$ , with $(\Xi (0),\alpha (0))$ the data of $r$ , with $\Xi _{i\cdot }(c)=\Xi _{i\cdot }$ and $\alpha _i(c)=\alpha _i$ for $i\notin I$ , with $\alpha _i(c)>0$ throughout the box, and with

\begin{equation} \alpha _i(c)+2\bigl (K_i(c)-\bar \Xi _i(c)\bigr )=\Delta A_i\qquad \text {for all }i\in I\text { and all }c, \label {eq:clock-preserving} \end{equation}

K_i(c)

and

\bar \Xi _i(c)

being the node quantities of (74) for the row

P_{i\cdot }

and the mark

\Xi (c)

. The variance-absorbing chart of declared directions

\Phi ^{(1)},\dots ,\Phi ^{(d)}

supported on

I\times \cA

\begin{equation} \Xi (c):=\Xi +\sum _{\kappa \le d}c_\kappa \Phi ^{(\kappa )},\qquad \alpha _i(c):=\alpha _i-2\bigl [(K_i-\bar \Xi _i)(c)-(K_i-\bar \Xi _i)(0)\bigr ], \label {eq:variance-absorbing} \end{equation}

on any certified closed box lying strictly inside the region on which every

\alpha _i(c)

is positive. Since

c\mapsto K_i(c)-\bar \Xi _i(c)

is the cumulant generating function of the centred mark composed with an affine map, it is convex and its maximum over a box is attained at a corner. Thus strict positivity on a proposed box is a finite corner check. One may compute the supremal admissible radius and then choose any smaller

\rho _c

; a largest closed box with strict positivity need not exist.

Theorem 10.9 (Exact declared spot call rows by a clock-preserving mark correction) . Let $r$ be a marked resolution, allowing also a marked time-inhomogeneous resolution with the clockless post- $P$ corrections of Theorem 10.6(iii); in that case $P$ below is the return-carrying move before the post- $P$ correction. An unmarked resolution is the case $\Xi \equiv 0$ , $\alpha =h\nu$ . Let $T>0$ be a grid date, and let $r_\kappa (S)=(S-K_\kappa )^+\wedge \bar M$ , $\kappa \le m$ , be capped spot call rows at $T$ with declared values $\tau _\kappa$ . Let $c\mapsto (\Xi (c),\alpha (c))$ be a clock-preserving mark chart on the step ending at $T$ , and let $r(c)$ be the marked time-inhomogeneous resolution that carries $(\Xi (c),\alpha (c))$ on that step and the data of $r$ on every other. Put

\begin{equation} F(c):=\bigl (\E _{r(c)}r_\kappa (S_T)-\E _rr_\kappa (S_T)\bigr )_{\kappa \le m},\qquad \varepsilon _\kappa :=\tau _\kappa -\E _rr_\kappa (S_T). \label {eq:spot-residual-map} \end{equation}

Then

F

is smooth on the box with

F(0)=0

. If

DF(0)

has rank

m

with least singular value

s>0

, if

\|D^2F\|\le L

on the ball

|c|\le \rho _c

, and if

|\varepsilon |\le \min \{s^2/(4L),\,s\rho _c/2\}

, with

s^2/(4L):=+\infty

when

L=0

, then there is exactly one

c

in the ball of radius

2|\varepsilon |/s

of the row space of

DF(0)

under which every declared spot row is exact. The atom chain, the clock, the budget field, the index and every index row of

r(c)

are those of

r

; the martingale and log-contract rows of Theorem 10.2 hold at every step of

r(c)

; and the corrected mark is in general not in gradient form.

The proof is given in Appendix 8.

Remark 10.10 (Level and shape) . The log-contract row ties the level of a spot smile to the clock: $\E [-2\log (S_T/S_0)]=\E A_T$ exactly, and a clock-preserving correction moves neither side. What such a correction reaches is the shape of the smile at fixed total variance, which at one step and near the money is close to one-dimensional: for three rows at strikes $0.95,1,1.05$ under a four-direction chart, the singular values of $DF(0)$ computed in Section 11.3 fall by a factor of about nine from each to the next. A clock-preserving chart cannot change the log-contract level. Thus a calibration target that also prescribes a different log-contract level at its maturity is not reached by this mark correction; that level has to come from the atom chain, which the index corrections of Theorems 7.4 and 7.8 move. This is the finite form of the consistency between spot and index smiles that a joint calibration has to respect, and it fixes the order of Corollary 10.11.

Corollary 10.11 (Exact finite joint panel at one resolution) . Let $r$ be a marked resolution with nondecreasing index map, and declare a finite panel: capped index call rows at fixings $T_1<\dots <T_J$ with $T_{j+1}-T_j\ge h$ , and capped spot call rows at grid maturities $T'_1<\dots <T'_{J'}$ , each maturity with a clock-preserving mark chart on the step ending at it. Assume all the hypotheses and finite certificates of Theorem 7.8 for the index rows, as imported in Theorem 10.6(iii), and perform that joint index correction. Then proceed through the spot maturities in increasing order. At each $T'_\iota$ , on the chain corrected by everything before, assume the chart, positive-variance box, full-rank Jacobian and Hessian bound of Theorem 10.9, and suppose that stage’s residual meets its threshold; apply that theorem before continuing. Then the resulting marked time-inhomogeneous resolution matches every declared row exactly. The dependency Jacobian is block lower triangular in this order wherever it is evaluated, because a spot correction moves no index row and a correction at a later maturity moves no earlier spot row. The diagonal blocks used in the staged inverse solves are the index block $\bar J=DR(0)$ of (53) at the index stage’s baseline and the spot blocks $DF(0)$ at their respective stage baselines. These blocks are evaluated at different intermediate chains, so they are not asserted to be the diagonal of one Jacobian evaluated at a single common base point.

The proof is given in Appendix 9.

Remark 10.12 (What is and is not claimed) . Corollary 10.11 is a statement about one resolution: a finite joint panel within the reach of the two corrections is exact there. The reach is certified, not assumed: the block-mass and slope conditions bound the index residual a block flow absorbs (Remark 7.9), while the least singular value, the certified Hessian bound and the positive-variance domain radius of the mark chart jointly bound the spot residual it absorbs (Theorem 10.9). Nothing is claimed about the corrected family: it is not promoted to a marked scheme, its corrected marks leave gradient form, and no convergence is asserted for an externally declared panel (Proposition 7.5 (c)). Which panels lie within reach at which resolution, and how the reach grows along a family, is the calibration question, and it is not addressed here.

11Extension experiments

These computations revisit the leverage and marked-correlation boundary tests of Sections 9 and 10. They do not verify the infinite-family range rule (60) or the weighted summability condition (84), and therefore are not presented as certified schemes. The final panel calculations are direct fixed-resolution solves: they address Sections 7.1 and 10.3; their row, domain and margin checks are exhaustive, but sampled Hessian maxima do not certify every hypothesis of the quantitative inverse theorem.

11.1A finite marked refinement experiment

Give the four base kernels the square-root mark (89) with $\varrho =-0.7$ . The mark-charge rule selects $\ell _m=0.999,1.060,1.117$ . Across all four resolutions, deterministic martingale and log-contract residuals are at most $1.8\cdot 10^{-15}$ and $5.6\cdot 10^{-17}$ . All four index maps are nondecreasing, as required by Theorem 10.6 (iii), although the finest clock-rate vector has a cap-induced reversal.

Nesting makes $d_m^\varphi$ and $d_m^{\sqrt {\mathsf s}}$ vanish numerically; the row-normalized $d_m^{\mathsf m}$ and $d_m^{\mathsf a}$ decay as shown in Table 2. Localized pairwise Lipschitz constants range over $1.81$ – $2.18$ and $10.68$ – $12.64$ , versus all-atom ranges $4.57$ – $6.57$ and $26.99$ – $42.29$ . This finite decay does not establish the infinite sum in (84), which also weights the base errors by the localized Lipschitz constants.

The paired diagnostics are in Table 2. Physical gaps range from one to about three standard errors, so four levels do not resolve a physical convergence rate; the martingale and log-contract rows, by contrast, hold before simulation.

11.2A finite leveraged range-sensitivity experiment

Take $\ell (x)=1.3-0.6\tanh ((x-x_0)/0.3)$ , with range $[0.7,1.9]$ and $L_x=2$ , and volatility caps $1.5,1.6,1.7$ . For sensitivity use

R_m=R_0\sqrt {\log (1/h_m)/\log (1/h_0)},

with $R_0=0.6$ and $1.0$ . These ranges fall far below the sufficient rule. Here $C_0=7.57$ for the variance kernels; Lemmas 9.3 and 5.1, with the Doob–Minkowski bounds, give $\beta '=9.03$ , $c_R=1.90$ and $m_4\le 2.0$ , so (61), for $\rho =2$ and $\delta _X=h^{3/2}$ , is met at $R\simeq 12.45,13.27,14.09$ . The required growth is $0.8$ per mesh halving, versus the leading $(\rho /c_R)\log 2=0.73$ . Its conservatism is visible because the proxy reaches a boundary on only $10^{-4}$ of path-steps already at $R=1$ . Odd proxy grids contain $x_0$ and use $\delta _X=R_m/\lceil R_m/h_m^{3/2}\rceil \le h_m^{3/2}$ . The state grids (volatility by proxy atoms) are $68\times 53,216\times 165,688\times 511$ for $R_0=0.6$ , and $68\times 87,216\times 273,688\times 849$ for $R_0=1.0$ .

Table 3. Finite leveraged refinement and range sensitivity,

\lambda =1

and

8\,000

paths. Each row compares the displayed coarse resolution

c

with the next finer one. States,

\delta _{X,c}

and

h_c

belong to the coarse chain. Proxy

_c

is the empirical

L^2

norm of the coarse chain’s pathwise maximum proxy error, including the terminal date; clamp

_c

is the fraction of coarse post-transition proxy states at a boundary.

X

and

U

are paired coarse–fine empirical

L^2

differences.

$R_0$	$m$	states $_c$	$\delta _{X,c}$	$h_c^{3/2}$	proxy $_c$	clamp $_c$	$X$ , $L^2$	$U$ , $L^2$
0.6	0	$68\times 53$	0.0231	0.0236	$3.3\cdot 10^{-2}$	$3.6\cdot 10^{-3}$	$3.7\cdot 10^{-2}$	$1.6\cdot 10^{-2}$
0.6	1	$216\times 165$	0.0083	0.0083	$2.0\cdot 10^{-2}$	$1.4\cdot 10^{-3}$	$2.7\cdot 10^{-2}$	$9.5\cdot 10^{-3}$
1.0	0	$68\times 87$	0.0233	0.0236	$2.3\cdot 10^{-2}$	$1.6\cdot 10^{-4}$	$3.7\cdot 10^{-2}$	$1.6\cdot 10^{-2}$
1.0	1	$216\times 273$	0.0083	0.0083	$1.1\cdot 10^{-2}$	$2.7\cdot 10^{-5}$	$2.7\cdot 10^{-2}$	$9.5\cdot 10^{-3}$

Across both sweeps, $X$ falls from $3.7\cdot 10^{-2}$ to $2.7\cdot 10^{-2}$ and $U$ from $1.6\cdot 10^{-2}$ to $9.5\cdot 10^{-3}$ . Increasing $R_0$ reduces proxy errors and boundary frequencies as shown in Table 3. These empirical sensitivities are not upper bounds for $\Q (E^R{}^c)$ , and the tested ranges do not meet (60) with the lemma’s constants.

11.3Finite panel solves at one resolution

Three direct finite solves, all at level $1$ ( $h=\Delta /2$ , $338$ atoms), use the iteration of Lemma 7.7 with the pseudo-inverse of the Jacobian at zero. Row values are finite sums, except that Fourier inversion for the level- $2$ four-step spot rows agrees with the exact level- $1$ two-step sum to $2.7\cdot 10^{-11}$ ; Jacobians use central differences. $\widehat L_{\rm samp}$ is the largest second derivative sampled at the centre, the $2d$ axes of an inner ball and six random points. It is not a certified bound on that ball, so its thresholds are counterfactual sample thresholds, although every direct solve and post-solve row residual is checked. These panel residuals are deterministic, not Monte Carlo.

Table 4. Diagnostics for the fixed-resolution panel solves at level

1

: the numerically evaluated least singular value

\widehat s

of the Jacobian at zero, sampled second-derivative maximum

\widehat L_{\rm samp}

, the ball’s radius

\bar \rho

(the flow-probability radius for index rows, the positive-variance half-width

\rho _c

for the mark chart), the resulting sample threshold, the Euclidean residual of the declared panel, and the residual after the solve. Because no certified global Hessian bound is supplied, the sample-threshold columns do not by themselves verify the quantitative inverse hypothesis.

panel	$\widehat s$	$\widehat L_{\rm samp}$	$\bar \rho$	$\widehat t_{\rm samp}$	$\|\varepsilon \|$	residual after solve
two fixings, index rows	$4.57\times 10^{-1}$	$9.30\times 10^{1}$	$5.05\times 10^{-5}$	$1.15\times 10^{-5}$	$8.35\times 10^{-6}$	$2.78\times 10^{-17}$
one spot row	$1.58\times 10^{-3}$	$6.24\times 10^{-3}$	$1.34\times 10^{-1}$	$9.97\times 10^{-5}$	$6.25\times 10^{-5}$	$5.61\times 10^{-16}$
joint panel, index part	$4.62\times 10^{-1}$	$5.59\times 10^{1}$	$9.00\times 10^{-5}$	$2.08\times 10^{-5}$	$5.36\times 10^{-6}$	$1.39\times 10^{-17}$
joint panel, spot part	$1.58\times 10^{-3}$	$6.24\times 10^{-3}$	$1.34\times 10^{-1}$	$9.98\times 10^{-5}$	$6.16\times 10^{-5}$	$6.70\times 10^{-16}$

Two fixings half a tenor apart. On the base scheme, fixings at $2\Delta$ and $2.5\Delta$ each carry three capped index call rows at strikes $0.162$ , $0.197$ , $0.239$ , placed at midpoints of neighbouring index values so that the index margin of Theorem 7.8 (i) holds by $1.5\cdot 10^{-3}$ against a largest displacement of $1.6\cdot 10^{-4}$ over the theorem’s full coordinatewise box. The declared panel raises the chain’s rows by one part in ten thousand. Coupling is material: $\|U\|=1.06$ versus diagonal-block norm $2.13$ , and $\|N\|=1.76$ ; the least singular value is $0.457$ . With the least move $\ell =\delta$ , the certified block masses lie in $[0.167,0.220]$ and the slopes in $[0.935,0.956]$ . The diagonal blocks match (54) to finite-difference accuracy. Three iterations give the residual in Table 4; $|z|=5.7\cdot 10^{-6}$ versus sample inverse radius $3.7\cdot 10^{-5}$ , and the largest signed perturbation is three percent of baseline. Solving fixing by fixing as in Theorem 7.4 leaves $7.1\cdot 10^{-7}$ on the first fixing’s rows. The level- $2$ chain’s own rows differ by $4.2\cdot 10^{-4}$ – $5.9\cdot 10^{-4}$ , or $36$ – $51$ times the vector sample threshold, so this experiment makes no reach claim one level down.

One spot row by a mark correction. On the marked chain at $\varrho =-0.7$ , the spot call at strike $1.05$ and maturity $\Delta$ , two steps, is corrected by the variance-absorbing chart on the second step, supported on the $38$ atoms that carry mass at least $10^{-4}$ there, with two directions: the local return scale times the first two orthonormal Hermite polynomials of the standardized atom increment, clipped at four standard deviations. The declared row comes from the same atom chain and clock with that step’s mark scaled to correlation $-0.6$ , the return variance absorbing the gap. Twelve iterations give the residual in Table 4. The coefficient norm is $0.042$ versus sample inverse radius $0.079$ ; return variances change by at most $19$ percent, clock increments by $4\cdot 10^{-19}$ , and martingale rows hold to $4\cdot 10^{-16}$ . The corrected mark leaves gradient form: row-to-row increment variation is five percent of its dispersion.

Level and shape. For the three rows at strikes $0.95,1,1.05$ and the four-direction chart, the singular values of $DF(0)$ are $2.5\cdot 10^{-3}$ , $3.5\cdot 10^{-4}$ , $2.8\cdot 10^{-5}$ . The level- $2$ chain’s own rows differ by $+0.6$ , $+1.2$ , $-0.4$ percent; the residual has norm $4.4\cdot 10^{-4}$ , almost all of it outside the leading direction. The clocks of the two levels differ, identifying a level component that a clock-preserving chart cannot alter; these local diagnostics make no reach or non-reach claim for the finite three-call panel. The rows of the chain with $\varrho =-0.5$ , whose clock also differs, have residual $7.5\cdot 10^{-4}$ with $2.7\cdot 10^{-4}$ outside the leading direction and $1.9\cdot 10^{-4}$ outside the leading two; the consistent panel above has $7.6\cdot 10^{-5}$ with $1.7\cdot 10^{-6}$ outside the leading two.

The joint panel. On the same marked chain, fixings at $\Delta /2$ and $\Delta$ , half a tenor apart, carry capped index call rows at strikes $0.178$ and $0.211$ , declared as the chain’s own rows raised by one part in ten thousand, and the spot row above at maturity $\Delta$ is declared as before. The index part is solved first, using clock rate $\mathsf a$ in the index map and return-free corrections. Here $\|U\|=0.67$ versus diagonal norm $1.45$ , and the residual is $1.4\cdot 10^{-17}$ . Index flows increase the spot row by $9.7\cdot 10^{-7}$ , exhibiting the lower-triangular law coupling of Corollary 10.11; the spot correction at the corrected chain is exact to $6.7\cdot 10^{-16}$ and leaves the index rows at $1.4\cdot 10^{-17}$ .

12Conclusion and scope

The construction above is a statement about finite objects and the family they form, and its reach is worth marking precisely. Target-free is not model-free: each resolution is a finite dynamic model, and no result here establishes empirical fit, hedging performance or sharp constants. Two limits of the demonstration deserve naming. The certificates are exhibited on a single family, generated by quantizing a square-root kernel (Lemma 2.7, Section 8), so the tables test the theory on an instance of exactly the analytic kind the theorems decline to name; Remark 2.9 states what those same certificates ask of a matrix produced by fitting, and supplying one belongs to a calibration procedure rather than to the certification. The panels solved here are the family’s own rows and those of a neighbouring chain; matching quoted panels, and deciding which of them lie within certified reach at a given resolution, is the question this framework is built to pose and hands on. The two extensions mark the architecture’s edges rather than widening them: state-dependent leverage retains the exact structural rows but can lose log-spot finite-mixture closure, its bounds are localized, and fixing-date physical transfer requires an exponential moment condition on $\lambda$ ; marked correlation retains Gaussian-mixture log-spot and common-budget rows, but its clock need not equal the predictable bracket and no linked density row is asserted (Remark 10.3), and the corrected marked family is not promoted to a marked scheme.

This paper certifies a refining family of finite linked markets on itself. The rows tying the two markets are identities of each resolution rather than constraints on a fitted law; the distance between neighbouring resolutions is assembled from their own matrices and bounds an adapted Wasserstein distance, so convergence becomes something a modeller checks rather than a target a modeller names; and a declared panel within certified reach is matched at a resolution without disturbing the limit the family already has. At no point does the construction require a continuous model to compare against. Three directions follow: successive calibrations should impose consistency during fitting rather than test it afterwards, polynomial-tail schemes may trade weaker rates for heavier tails, and whether every continuously certified market arises as the limit of such finite schemes is the converse problem.

AI-use disclosure. The author used Anthropic Claude Code and OpenAI Codex as interactive research and writing tools. They assisted with exploratory discussion, testing and refinement of ideas, literature and source organization, code development and verification, mathematical error checking, and editorial revision.

AThe square-root construction

Proof of Lemma 2.7 . For integer $d_B$ , the CIR process is the squared radius of the $d_B$ -dimensional Ornstein–Uhlenbeck process

\begin{equation} d\mathbf Z_t=-\frac \kappa 2\mathbf Z_t\,dt+\frac \sigma 2\,d\mathbf B_t, \qquad v_t=|\mathbf Z_t|^2 . \label {eq:cir-ou} \end{equation}

Contraction. Start two copies at vectors on the same ray whose radii differ by $\delta =\delta _m$ , and give them the same Gaussian increment. The reverse triangle inequality couples their time- $h$ radii at distance at most $s=e^{-\kappa h/2}\delta$ . Radial OU kernels are stochastically ordered in the initial radius. Their ordered quantile coupling therefore also has distance at most $s$ : stochastic order gives the sign of the quantile difference, while the preceding coupling gives its $W_\infty$ bound.

The radial transition density $f_{h,a}$ is a scaled noncentral $\chi$ density, and its total variation is bounded as follows. In a frame with $\mathbf z=ae_1$ , (93) gives $\mathbf Z_h=\tilde ae_1+s_hG$ with $\tilde a:=e^{-\kappa h/2}a$ and $s_h^2=\sigma ^2(1-e^{-\kappa h})/(4\kappa )$ , so $|\mathbf Z_h|=s_h\,|\mu e_1+G|$ with $\mu :=\tilde a/s_h$ : the radius is $s_h$ times a noncentral $\chi$ variable with $d_B$ degrees of freedom and noncentrality $\mu$ . Its density is the Poisson mixture

g_{d_B,\mu }=\sum _{j\ge 0}e^{-\mu ^2/2}\frac {(\mu ^2/2)^j}{j!}\,\chi _{d_B+2j}, \qquad \chi _k(u):=\frac {2^{1-k/2}}{\Gamma (k/2)}\,u^{k-1}e^{-u^2/2}\quad (u\ge 0),

of central

\chi

densities, the radial form of the Poisson-mixture representation of the noncentral

\chi ^2

law. Total variation is a seminorm and the weights sum to one, so

\operatorname {Var}(g_{d_B,\mu })\le \sup _{k\ge d_B}\operatorname {Var}(\chi _k)

. For

k\ge 2

the density

\chi _k

increases on

[0,\sqrt {k-1}]

and decreases after it, so

\operatorname {Var}(\chi _k)=2\chi _k(\sqrt {k-1})

, and Stirling’s lower bound

\Gamma (x)\ge \sqrt {2\pi }\,x^{x-1/2}e^{-x}

gives

\chi _k(\sqrt {k-1}) =\frac {2^{1-k/2}(k-1)^{(k-1)/2}e^{-(k-1)/2}}{\Gamma (k/2)} \le \frac {e^{1/2}}{\sqrt \pi }\Bigl (\frac {k-1}{k}\Bigr )^{(k-1)/2} \le \frac {e^{1/2}}{\sqrt \pi }.

Hence

\operatorname {Var}(g_{d_B,\mu })\le 2e^{1/2}/\sqrt \pi

for every

\mu \ge 0

and every

d_B\ge 2

, uniformly. Scaling a density by

s_h

divides its total variation by

s_h

, and

s_h^2\ge \sigma ^2he^{-\kappa h}/4

, so for

\kappa h\le 1

\begin{equation} \sup _{a\ge 0}\operatorname {Var}(f_{h,a}) :=\sup _{a\ge 0}\int _0^\infty |\partial _y f_{h,a}(y)|\,dy =\sup _{\mu \ge 0}\frac {\operatorname {Var}(g_{d_B,\mu })}{s_h} \le \frac {4e}{\sigma \sqrt \pi }\,h^{-1/2}. \label {eq:cir-density-tv} \end{equation}

For a density

f

of bounded variation and grid boundaries separated by

\delta

, the elementary shifted-Riemann-sum bound is

\begin{equation} \sum _j\int _{b_j-s}^{b_j}f(y)\,dy \le \frac {s}{\delta }\{1+\delta \operatorname {Var}(f)\}, \qquad 0\le s\le \delta . \label {eq:boundary-riemann} \end{equation}

Apply this to the exact ordered coupling and then the monotone quantizer. Its quantized outputs differ by one grid step only if an interval of length

s

crosses a boundary. Hence

W_\rho ^\rho \{P_m(a+\delta ,\cdot ),P_m(a,\cdot )\} \le e^{-\kappa h/2} \{1+\delta \operatorname {Var}(f_{h,a})\}\delta ^\rho .

Because

\delta _mh_m^{-1/2}=o(h_m)

, taking the

\rho

th root proves (a). Stochastic monotonicity of the exact radial kernel and of

q_m

proves the order part of (C1).

Lyapunov and diffusive-move bounds. At the lower exterior cell the safe one-sided estimate is $q_m(y)\le y+\delta _m$ ; at the upper exterior cell quantization only decreases $y$ . Consequently

V_\beta (q_m(y)) \le e^{\beta (2Y_m\delta _m+\delta _m^2)}V_\beta (y).

The exact CIR affine transform gives

K_hV_\beta (a)\le e^{C_\beta h}V_\beta (a)

when

\beta <2\kappa /\sigma ^2

. Since

Y_m\delta _m/h_m\to 0

, decreasing

h_0

if necessary gives one common

c_\beta

in (C2).

For (C4), (93) gives, for $|\mathbf z|=a$ ,

\begin{equation} \mathbf Z_h=e^{-\kappa h/2}\mathbf z+s_hG,\qquad s_h^2=\frac {\sigma ^2(1-e^{-\kappa h})}{4\kappa } \le \frac {\sigma ^2h}{4},\qquad G\sim N(0,I_{d_B}). \label {eq:cir-ou-step} \end{equation}

Upper clipping cannot increase the distance from the initial atom, while an interior or lower-cell rounding costs at most

\delta _m

. Uniformly for

a\le Y_m

\begin{equation} |q_m(|\mathbf Z_h|)-a| \le (1-e^{-\kappa h/2})Y_m+s_h|G|+\delta _m . \label {eq:cir-ou-increment} \end{equation}

Put

r_m=\{(1-e^{-\kappa h_m/2})Y_m+\delta _m\}/\sqrt {h_m}

. The cap and spacing conditions give

r_m\to 0

. Thus

\E \exp \!\left \{\vartheta _* \frac {|q_m(|\mathbf Z_h|)-a|^2}{h}\right \} \le e^{2\vartheta _*r_m^2} \E e^{(\vartheta _*\sigma ^2/2)|G|^2} =e^{2\vartheta _*r_m^2} (1-\vartheta _*\sigma ^2)^{-d_B/2}.

The explicit value (18) makes the last expression at most

2

after decreasing

h_0

, proving (C4).

Two-scale defect and cap events. Couple one exact step with two exact half-steps of (93). At the intermediate fine time, keep the direction and quantize the radius, then use the same second-half Gaussian increment. Away from exterior cells, the coarse endpoint rounding, the propagated intermediate rounding and the fine endpoint rounding cost at most $\delta _m+2\delta _{m+1}$ .

Let $E_m$ be the union of the coarse endpoint cap event and the two fine cap events. Set

L_m:=\log \frac {4\Xi _m}{\delta _{m+1}},\qquad g_m:=Y_m-\ell _m .

Before a cap is hit, the intermediate rounding moves the unrounded full-step radius by at most

\delta _{m+1}

. The boundary of an exterior cap cell adds at most another

\delta _m

, so

2\delta _m

is a common conservative cushion for all three events. For

G\sim N(0,I_{d_B})

\begin{equation} \Q \{|G|\ge \sqrt {d_B}+\sqrt {2x}\}\le e^{-x}. \label {eq:gaussian-radius-tail} \end{equation}

Because

L_m\to \infty

\delta _m/\sqrt {h_m}\to 0

, and the cap condition has the fixed slack

1+\eta

, decreasing

h_0

makes, simultaneously for every

m

\begin{equation} \frac {g_m-2\delta _m}{\sigma \sqrt {h_m}/2} \ge \sqrt {d_B}+\sqrt {2\rho L_m}. \label {eq:cap-tail-threshold} \end{equation}

Apply (98) to the half-step and full-step Gaussian increments. Mean reversion is inward, stochastic monotonicity reduces the worst initial radius to

\ell _m

, and a union bound gives

\sup _{a\le \ell _m}\Q _a(E_m) \le 3e^{-\rho L_m} =3\left (\frac {\delta _{m+1}}{4\Xi _m}\right )^\rho \le \left (\frac {\delta _{m+1}}{\Xi _m}\right )^\rho .

E_m

the discrepancy is at most

\Xi _m

; on its complement it is at most

\delta _m+2\delta _{m+1}

. Minkowski’s inequality therefore yields

d_{\ell _m}(r_m,r_{m+1}) \le \delta _m+3\delta _{m+1},

which is (19).

Finally the lattices are nested, and the common initial atom gives $\iota _{\rho ,m}=\zeta _m=0$ . Condition (16) makes the excursion charge at most $\delta _{m+1}/2$ , while (a) gives $1-c_{m+1}^2\ge \kappa h_{m+1}/(2\rho )$ . Substitution in (14) proves $\varepsilon _m=O(\rho \delta _m/h_m)=O(h_m^{1/2})$ and completes the proof.∎

BProofs for exactification, leverage and correlation

This appendix collects the longer technical arguments used by the exactification and boundary-test sections. The statements, hypotheses and quantitative constants remain in the body; only their proofs are deferred here.

B.1Shared quantitative-inverse argument

B.2Coupled index exactification

Proof of Theorem 7.8 . On the box $|q_{j,\iota }|\le \bar p_j/2$ , so the flows are stochastic by the computation in the proof of Theorem 7.4. Each correction matrix is affine in its block $z_j$ and occurs at most once in any ordered product. Thus the law of the corrected chain, the pre-correction law at each fixing and (50) are polynomials, multi-affine in the scalar coordinates of $z$ . The first term of (50) is $\nu (a)>0$ , so $\psi ^z_j$ is smooth in $z$ . By (i) each row $r_{j,\iota }$ is affine on the range of $\psi ^z_j(a)$ over the box, for every atom $a$ . The strict margins and compactness give the same property on a neighbourhood of the box. Hence $R$ is twice continuously differentiable there with $R(0)=0$ , and Lemma 7.7 applies with $d=m=K$ , $\rho =\bar \rho$ and the certificate (ii), the ball of radius $2|\varepsilon |/\bar s\le \bar \rho$ lying in the box; $R(z)=\varepsilon$ is the exactness of every row, and the flows are supported on the blocks and their shifts. For (53): the pre-correction law entering $C^{(j)}$ at $T_j$ involves only the correction kernels at earlier fixings, hence $z_{j'}$ for $j'<j$ ; the index map at $T_j$ involves the steps $k_j,\dots ,k_j+m_\Delta -2$ , which contain the corrected step $k_{j'}-1$ exactly when $j'\in W_j$ ; and the derivative with respect to $z_{j,\iota }$ at fixed law and index map is the row change of the proof of Theorem 7.4, namely $\ell _j\bigl [\mu ^{(j)}_0(I_{j,\iota })s^+_{j,\iota }(0)+(\bar p^-_{j,\iota }/\bar p_j)\mu ^{(j)}_0(I_{j,\iota }+\ell _j)s^-_{j,\iota }(0)\bigr ]\1\{\iota '\le \iota \}$ per unit of $q_{j,\iota }$ , times $\partial q_{j,\iota }/\partial z_{j,\iota }=1/(2\ell _jm_{j,\iota })$ ; under the uncorrected chain the definition of $\bar p^-_{j,\iota }$ makes the two terms equal and the entry $s^+_{j,\iota }$ . If every $W_j$ is empty, $\bar J=\widetilde M+N$ is block lower triangular, and a triangular system with invertible diagonal blocks is solved by forward substitution. Each block uses the same triangular row-flow calculation as Theorem 7.4, evaluated at the pre-correction law left by the earlier fixings.∎

B.3Leverage estimates

Proof of Lemma 9.3 . Write $z_k:=\hat x_k+X_{k+1}-X_k$ , so that $\hat x_{k+1}=q_X(z_k)$ and

\hat x_{k+1}-X_{k+1}=\bigl [q_X(z_k)-z_k\bigr ]+\bigl (\hat x_k-X_k\bigr ).

E^R

the point

z_k

lies within the grid’s range, since

|z_k-x_0|\le \pi _k+|X_{k+1}-x_0|\le H\delta _X/(2h)+R-H\delta _X/(2h)\le R

by induction, so

q_X

does not clamp and

|q_X(z_k)-z_k|\le \delta _X/2

; the displayed identity then gives

\pi _{k+1}\le \pi _k+\delta _X/2

and

\pi _0=0

. Off

E^R

the crude bound

\pi _k\le R+\max _k|X_k-x_0|

holds because

\hat x_k

stays in the grid. Splitting on

E^R

and applying Hölder gives (58). The moment bound is

|X_k-x_0|\le \tfrac 12A_{t_k}+|M_k|

with

M

the return martingale, whose bracket is

A_{t_k}\le \ell _+^2H\max _k\nu _{y_k}

; apply Doob and Burkholder–Davis–Gundy and Lemma 5.1.

For (59), write $X_k-x_0=-\tfrac 12A_{t_k}+M_k$ with $M_k:=\sum _{i<k}\sigma _i\Delta W_i$ . Since $\sigma _i$ is $\cF _i$ -measurable and $\Delta W_i\sim N(0,h)$ is independent of $\cF _i$ , for every real $\lambda$ the process $Z_k:=\exp (\lambda M_k-\tfrac 12\lambda ^2A_{t_k})$ is a nonnegative martingale with $Z_0=1$ . Fix $x>0$ and $\lambda :=R/x$ . On $\{\max _{k\le n}M_k\ge R\}\cap \{A_{t_n}\le x\}$ , at the first $k$ with $M_k\ge R$ one has $Z_k\ge e^{\lambda R-\lambda ^2x/2}=e^{R^2/(2x)}$ , so Doob’s maximal inequality gives $\Q (\max _kM_k\ge R,\ A_{t_n}\le x)\le e^{-R^2/(2x)}$ , and the same for $-M$ . Since $A_{t_n}\le \ell _+^2H\max _k\nu _{y_k}$ , Lemma 5.1 gives $\Q (A_{t_n}>x)\le C_0e^{-\beta 'x}$ ; the choice $x:=R/\sqrt {2\beta '}$ makes the two exponents equal and yields $\Q (\max _k|M_k|\ge R)\le (C_0+2)e^{-R\sqrt {\beta '/2}}$ . Finally $|X_k-x_0|\le \tfrac 12A_{t_k}+|M_k|$ , so for $\theta \in (0,1)$

\begin{align*} \Q \Bigl (\max _k|X_k-x_0|>R\Bigr ) &\le \Q \Bigl (\tfrac 12\ell _+^2H\max _k\nu _{y_k}>\theta R\Bigr ) +\Q \Bigl (\max _k|M_k|>(1-\theta )R\Bigr )\\ &\le C_0e^{-2\theta \beta 'R}+(C_0+2)e^{-(1-\theta )R\sqrt {\beta '/2}}, \end{align*}

and the $\theta$ with $2\theta \beta '=(1-\theta )\sqrt {\beta '/2}$ makes both exponents equal to $c_RR$ , which is (59). For (61), insert (59) into (60) and take logarithms; with $\delta _X\asymp h^{3/2}$ , $\log (2h/(H\delta _X))=\tfrac 12\log (1/h)+O(1)$ , and along $R_m\asymp \log (1/h_m)$ the term $\log (R_m+m_{2\rho })$ is $O(\log \log (1/h_m))$ , which gives the stated growth.∎

Proof of Lemma 9.4 . Let

\tau :=\inf \Bigl \{j\in \{0,\ldots ,2n-1\}: h'\!\sum _{i\le j}\bigl ((y'_i)^2+y_{k(i)}^2\bigr )>K \ \text {or}\ \nu _{y'_j}\vee \nu _{y_{k(j)}}>1/h'\Bigr \},

with the convention

\inf \varnothing =2n

. It is a stopping time of the joint filtration, and

E_K\subseteq \{\tau =2n\}

, so it suffices to bound

D_{j\wedge \tau }

. For

i<\tau

\begin{equation} h'(y'_i)^2\le 1,\qquad h'y_{k(i)}^2\le 1,\qquad h'\!\!\sum _{i'<i}\bigl ((y'_{i'})^2+y_{k(i')}^2\bigr )\le K. \label {eq:lsv-stopped} \end{equation}

One step. Let $\mathcal H'_j$ be generated by the two complete autonomous atom paths and all their coupling randomizations, together with $W_s$ , $s\le t'_j$ . Then $D_j$ , $\Delta _j$ and $b_j$ are $\mathcal H'_j$ -measurable, while $\Delta W'_j$ is independent of $\mathcal H'_j$ and is $N(0,h')$ . Write $u_j:=L_xy'_j$ and $\zeta _j:=\Delta _j+b_j\Delta W'_j$ . For $q\ge 2$ there is a constant with $|a+z|^q\le |a|^q+q|a|^{q-2}az+C_q(|a|^{q-2}z^2+|z|^q)$ for all real $a,z$ . Conditionally on $\mathcal H'_j$ the increment $\Delta W'_j$ is centred with variance $h'$ and independent of $(\Delta _j,b_j)$ , so

\begin{equation} \E \bigl [|D_{j+1}|^q\mid \mathcal H'_j\bigr ] \le |D_j|^q+q|D_j|^{q-1}|\Delta _j| +C_q\Bigl (|D_j|^{q-2}\bigl (\Delta _j^2+h'b_j^2\bigr )+|\Delta _j|^q+h'^{q/2}|b_j|^q\Bigr ). \label {eq:lsv-onestep} \end{equation}

Nothing is discarded: the second-order and

q

-th order remainders in

\Delta _j

are kept and absorbed below. Since

\bar \sigma _j+\sigma '_j\le \ell _+(y_{k(j)}+y'_j)

and

\bar \sigma _j^2-(\sigma '_j)^2=b_j(\bar \sigma _j+\sigma '_j)

|\Delta _j|\le \tfrac 12h'\ell _+\bigl (y_{k(j)}+y'_j\bigr )|b_j| ,

and (100) gives

h'(y_{k(j)}+y'_j)^2\le 4

for

j<\tau

, whence

\Delta _j^2\le h'\ell _+^2h'(y_{k(j)}+y'_j)^2b_j^2/4\le h'\ell _+^2b_j^2

and

|\Delta _j|^q\le (\ell _+/2)^qh'^{q/2}\bigl (h'(y_{k(j)}+y'_j)^2\bigr )^{q/2}|b_j|^q \le (2\ell _+)^qh'^{q/2}|b_j|^q

. So every term of (101) beyond the first is at most a constant multiple of

h'\ell _+\bigl (y_{k(j)}+y'_j\bigr )|D_j|^{q-1}|b_j| \ +\ h'(1+\ell _+^2)|D_j|^{q-2}b_j^2 \ +\ (1+2\ell _+)^q\,h'^{q/2}|b_j|^q .

Insert

|b_j|\le u_j|D_j|+g_j

. For the mixed part of the first term, first use

(y_{k(j)}+y'_j)|D_j|^{q-1}g_j \le \tfrac 12(y_{k(j)}+y'_j)^2|D_j|^q +\tfrac 12|D_j|^{q-2}g_j^2,

and then use Young’s inequality

|D|^{q-2}g^2\le \frac {q-2}q|D|^q+\frac 2qg^q

; the other mixed terms are handled by the same inequality and by

|D|^{q-1}g\le \frac {q-1}q|D|^q+\frac 1qg^q

. Thus each term splits into a multiple of

|D_j|^q

and a multiple of

g_j^q

. Apart from a deterministic coefficient, the three state-dependent coefficients of

|D_j|^q

are handled by (100):

h'\ell _+(y_{k(j)}+y'_j)u_j\le h'\ell _+L_x\bigl (y_{k(j)}^2+\tfrac 32(y'_j)^2\bigr )

2ab\le a^2+b^2

;

h'(1+\ell _+^2)u_j^2=h'(1+\ell _+^2)L_x^2(y'_j)^2

; and

h'^{q/2}u_j^q=h'\,L_x^q\,(y'_j)^2\,\bigl (h'(y'_j)^2\bigr )^{q/2-1}\le h'L_x^q(y'_j)^2 ,

which is where the state bound

\nu _{y'}\le 1/h'

is used and is the only place it is needed. Also

h'^{q/2}g_j^q\le (1\vee H^{q/2-1})h'g_j^q

, since

h'\le H

. Collecting and absorbing this fixed-horizon factor into

C_{q,H}

, for

j<\tau

\E \bigl [|D_{j+1}|^q\mid \mathcal H'_j\bigr ] \le \bigl (1+h'(C_q+\lambda _j)\bigr )|D_j|^q+C_{q,H}h'g_j^q, \qquad \lambda _j:=\kappa _q\bigl ((y'_j)^2+y_{k(j)}^2\bigr ),

with

\kappa _q

as in (64).

Unrolling. Put $V_j:=|D_{j\wedge \tau }|^q e^{-C_qt'_{j\wedge \tau }-\Gamma _{j\wedge \tau }}$ . Since $e^{-\Gamma _{j+1}}=e^{-\Gamma _j}e^{-h'\lambda _j}$ and $e^{-x}(1+x)\le 1$ , the display above gives $\E [V_{j+1}\mid \mathcal H'_j]\le V_j+C_{q,H}h'g_j^q$ on $\{\tau >j\}$ , and $V_{j+1}=V_j$ on $\{\tau \le j\}$ . As $D_0=0$ ,

\E V_j\ \le \ C_{q,H}\sum _{i<2n}h'\,\E g_i^q\ \le \ C_{q,H}\,H\,\max _{i<2n}\|g_i\|_{L^q}^q \qquad (j\le 2n).

No maximal inequality is invoked, and none is available for

V

L^1

. On

E_K

one has

\tau =2n

and

\Gamma _{2n}\le \kappa _qK

, so

|D_j|^q\1_{E_K}\le e^{C_qH+\kappa _qK}V_j

and (66) follows by taking

q

-th roots. For the tail,

h'\sum _{j<2n}((y'_j)^2+y_{k(j)}^2)\le H(\max _j\nu _{y'_j}+\max _k\nu _{y_k})

, so Lemma 5.1 bounds

\Q (E_K^c)

as stated.

The inhomogeneity. In (63), $e_{k(j)}$ has $L^q$ norm at most $\varepsilon ^{(q)}_m$ by Lemma 4.4; $|y'_{2k(j)}-y'_j|$ is at most one fine step of the finer chain, so $\|\max _j|y'_{2k(j)}-y'_j|\|_{L^q}\le C\Lambda _{2n}$ by Lemma 5.2 (a); the coarse cell oscillation obeys $\|\max _ko_k\|_{L^{2q}}\le C\Lambda _n$ by Lemma 5.2 (c); the proxy errors are $O(H\delta _X/(2h))=O(\sqrt h)$ in $L^{2q}$ by Lemma 9.3 under $\delta _X\le h^{3/2}$ ; and $L_thy'_j$ is $O(h)$ . Multiplying by $y'_j$ where required and using Lemma 5.1 with Cauchy–Schwarz gives (67).∎

Proof of Theorem 9.5 . By Theorem 9.2 and Lemma 4.2 both chains are co-immersed in the joint filtration, so Proposition 4.5 applies with $Y:=A_w-A_u$ and $Y':=A'_w-A'_u$ and reduces (68) to a bound on $\|Y-Y'\|_{L^\rho }$ . Over one fine step the two clocks differ by $h'|\bar \sigma _j^2-(\sigma '_j)^2|\le h'\ell _+(y_{k(j)}+y'_j)|b_j|$ with $b_j$ as in (62), and $|b_j|\le L_xy'_j|D_j|+g_j$ . Split each summand on $E_K$ and its complement. On $E_K$ , Cauchy–Schwarz against Lemma 5.1 and then (66) and (67) at $q=2\rho$ bound the $L^\rho$ norm of the $j$ -th summand by $h'C_K(\varepsilon ^{(2\rho )}_m+\Lambda _{2n})$ ; there are $(w-u)/h'$ of them. Off $E_K$ the crude bound $|Y-Y'|\le (w-u)(\max _k\sigma _k^2+\max _j(\sigma '_j)^2)$ and Hölder give the second term. The index divides by $\Delta$ ; the floor is $U\ge \ell _-^2u_-$ because every $\sigma ^2\ge \ell _-^2\min _i\nu _i$ ; and the spot statement is Lemma 9.4 at $j=2n$ .

For (70), on $E_K$ use $|b_j|\le L_xy'_j|D_j|+g_j$ , Cauchy–Schwarz, and Lemma 9.4 at $q=8$ :

\|b_j\1_{E_K}\|_{L^4} \le L_x\|y'_j\|_{L^8}\|D_j\1_{E_K}\|_{L^8} +\|g_j\|_{L^4} \le C_H e^{\kappa _8K/8} (\varepsilon _m^{(8)}+\Lambda _{2n}).

E_K^c

, the leverage bounds give

|b_j|\le \ell _+(y_{k(j)}+y'_j)

, so Hölder and Lemma 5.1 bound its

L^4

norm by

C\Q (E_K^c)^{1/8}

. The same split for

D_{2n}

uses the order-

8

localized estimate on

E_K

and

\|D_{2n}\1_{E_K^c}\|_{L^2}\le \|D_{2n}\|_{L^4}\Q (E_K^c)^{1/4}

off it; the uniform

L^4

bound follows from the bounded leverage and Lemma 5.1. This proves (70).

The density-ratio calculation of Proposition 5.5, now with integrand $b$ , gives

H(\Pp \,|\,\Pp ')=\tfrac 12\lambda ^2 \E _{\Pp }\sum _{j<2n}h'b_j^2.

By Cauchy–Schwarz and Minkowski,

\E _{\Pp }\sum _jh'b_j^2 \le \|L\|_{L^2(\Q )} \left \|\sum _jh'b_j^2\right \|_{L^2(\Q )} \le \mathcal L_2\sum _jh'\|b_j\|_{L^4(\Q )}^2,

which together with (70) and Pinsker proves (71). Since

L

is measurable for the coarse chain’s terminal filtration and

L'

for the fine chain’s terminal filtration, Bayes’ formula shows that their respective

\Q

-immersions in the joint filtration are preserved under

\Pp

and

\Pp '

. The conditional-readout decomposition in the same proposition therefore applies and bounds its payoff term by

L_f\E _{\Pp }|D_{2n}|\le L_f\mathcal L_2\|D_{2n}\|_{L^2(\Q )}

and its change-of-measure term by

2M|\lambda |(\E _{\Pp }\sum _jh'b_j^2)^{1/2}

, proving (72). Thus the

L^4

estimate for

b

, not merely the density moment, is the reason for the extra order-

8

hypothesis.

For (73) itself, write $L^2=\mathcal E(4M)^{1/2}e^{3\langle M\rangle }$ with $M:=\lambda \int \sigma \,dW$ and apply Cauchy–Schwarz.∎

B.4Marked-correlation estimates and clock-preserving spot rows

Proof of Lemma 10.5 . Apply (81) at both resolutions and subtract. On $E_\ell$ , each endpoint potential difference is at most $d_m^\varphi +L_m^\varphi (e_k+\zeta _m)$ , where $e_k:=|y_k-y'_{2k}|$ ; off $E_\ell$ it is at most $2R_m^\varphi$ . Taking $L^\rho$ norms, using Lemma 4.4 and charging $E_\ell ^c$ by (83) gives the first line of (87), for both endpoints together.

The drift sums satisfy

\Bigl |h\sum _{k<n}\mathsf m_{y_k}-h'\sum _{j<2n}\mathsf m'_{y'_j}\Bigr | \le h\sum _{k<n}\Bigl (\bigl |\mathsf m_{y_k}-\mathsf m'_{y'_{2k}}\bigr | +\tfrac 12\bigl |\mathsf m'_{y'_{2k+1}}-\mathsf m'_{y'_{2k}}\bigr |\Bigr ),

E_\ell

the first difference in each summand is at most

d_m^{\mathsf m}+L_m^{\mathsf m}(e_k+\zeta _m)

and the second at most

L_m^{\mathsf m}J'

. Off

E_\ell

the whole time average is bounded by a constant times

HR_m^{\mathsf m}

. Minkowski, the coupling recursion and the mark charge therefore give the second line of (87).

The martingale difference is a sum of centred increments with predictable coefficients. Put $c_j:=\sqrt {\mathsf s_{y_{k(j)}}}-\sqrt {\mathsf s'_{y'_j}}$ . Burkholder–Davis–Gundy bounds its $L^\rho$ norm by

C_\rho \left \|\left (\sum _{j<2n}h'c_j^2\right )^{1/2}\right \|_{L^\rho }.

E_\ell

each coefficient is at most

d_m^{\sqrt {\mathsf s}}+L_m^{\sqrt {\mathsf s}} \bigl (e_{k(j)}+|y'_{2k(j)}-y'_j|+\zeta _m\bigr ).

Since

\rho \ge 2

, Minkowski’s inequality in

L^{\rho /2}

, applied directly to the quadratic variation, gives

\begin{align*} \left \|\left (\sum _{j<2n}h'c_j^2\right )^{1/2}\1_{E_\ell }\right \|_{L^\rho } &\le \left (\sum _{j<2n}h'\|c_j\1_{E_\ell }\|_{L^\rho }^2\right )^{1/2}\\ &\le C\sqrt H\Bigl [d_m^{\sqrt {\mathsf s}} +L_m^{\sqrt {\mathsf s}}\bigl (\varepsilon _m^{(\rho )} +\Lambda _{2n}+\zeta _m\bigr )\Bigr ]. \end{align*}

Here the coupling error is used only through the pointwise estimate $\|e_k\|_{L^\rho }\le \varepsilon _m^{(\rho )}$ ; no estimate of $\max _k e_k$ is invoked. Off $E_\ell$ the quadratic variation is at most $4H(R_m^{\sqrt {\mathsf s}})^2$ , and its $L^\rho$ contribution is bounded by $2\sqrt H\,\chi _m^{\sqrt {\mathsf s}}$ . This gives the third line of (87). Finally the fine half-step term is bounded by $C\Lambda _{2n}$ by Lemma 5.2 (a), and $\zeta _m\le (1-c_{m+1}^2)\varepsilon _m^{(\rho )}$ by (14). These estimates also show explicitly that no Brownian tail or high-atom event has been discarded.∎

Proof of Theorem 10.6 . (i) The atom chain and its kernel are those of Definition 2.1, so Lemma 4.4 holds unchanged, and by (76) the budget field is $B(t_k;u,w)=h\sum _m(P^{m-k}\mathsf a)_{y_k}$ , which is the base case with $\mathsf a$ in place of $\nu$ . Proposition 4.5 therefore applies with $Y=A_w-A_u$ , and over one coarse step the clocks differ by $h(|\mathsf a_{y_k}-\mathsf a'_{y'_{2k}}|+\tfrac 12|\mathsf a'_{y'_{2k+1}}-\mathsf a'_{y'_{2k}}|)$ , on $E_\ell$ by $d_m^{\mathsf a}+L_m^{\mathsf a}(\varepsilon ^{(\rho )}_m+\zeta _m)$ and $L_m^{\mathsf a}J'$ , respectively. Off $E_\ell$ their time average is bounded by a constant times $R_m^{\mathsf a}$ , and (83) pays for that event. Sum over the window and use Lemma 5.2 (a) for $J'$ . The floor holds because $\mathsf a\ge \mathsf s>0$ and $P^m\mathsf a$ is a convex combination of the entries of $\mathsf a$ . (ii) is Lemma 10.5; for the physical layer, the separately chosen densities are stochastic exponentials against the same $W$ with atom-measurable integrands $\lambda \sqrt {\mathsf s}$ , so the proof of Proposition 5.5 applies verbatim with that integrand. (iii) The correction acts on the atom chain, and every quantity in Theorems 7.4 and 7.8 is computed from $P$ , the index map and the relevant pre-correction laws. The hypotheses above supply, rather than infer from monotonicity, every block, mass, slope, capacity and inverse certificate used by the applicable theorem; monotonicity only supplies the ordering of the index map. Condition (C1) together with a nondecreasing $\mathsf a$ is one sufficient way to check that ordering. Under the tenor condition no correction falls inside a window, so $\psi _n$ is the corrected chain’s index map at every fixing; otherwise that map is (50) with $\mathsf a$ in place of $\nu$ , and the clock rate at a corrected step is still $\mathsf a$ , because the return and the clock of that step are the $P$ -move’s. The identities of Theorem 10.2 at a corrected step are those of the $P$ -move. Conditional on the atom $i$ at the start of the step, first sample the pre-correction atom $j$ from $P(i,\cdot )$ and then the return from $N(\mu _{ij},\alpha _i)$ . Averaging over $j$ gives the martingale row and the log-contract row with $\Delta A_i$ exactly as in (77); the extra transition, which carries neither return nor clock, leaves both unchanged.∎

Proof of Theorem 10.9 . Write $\widetilde y_{k+1}$ for the endpoint of the return-carrying $P$ -move; it equals $y_{k+1}$ on an ordinary step and is the recorded pre-correction atom when a clockless post- $P$ correction is present. Given the complete atom path, including these recorded atoms, the log-spot at $T$ is Gaussian with mean $\sum _{k<k_T}\mu _{y_k\widetilde y_{k+1}}(c)$ and variance $\sum _{k<k_T}\alpha _{y_k}(c)$ , only the step ending at $T$ depending on $c$ . A capped call is the difference of two Black–Scholes functions of that mean and variance, smooth while the variance is positive, and $\mu _{ij}(c)=\Xi _{ij}(c)-K_i(c)-\tfrac 12\alpha _i(c)$ is smooth in $c$ ; so $F$ is a finite sum of smooth functions of $c$ , and Lemma 7.7 gives the solution. The chart leaves $P$ and, by (90), every $\Delta A_i$ unchanged, and by (76) the clock rate $\mathsf a$ with them; the budget field, the index and every index row are functions of $P$ and $\mathsf a$ alone. The node correction gives $\sum _jP_{ij}e^{\mu _{ij}(c)+\alpha _i(c)/2}=1$ , the martingale row, and $-2\sum _jP_{ij}\mu _{ij}(c)=\alpha _i(c)+2(K_i(c)-\bar \Xi _i(c))=\Delta A_i$ , the log-contract row; on the other steps nothing changed. Gradient form requires $\Xi _{ij}-\Xi _{i'j}$ to be independent of $j$ ; a nonconstant chart supported on a single atom, or increments that vary with $j$ differently at two atoms, generally break it.∎

Proof of Corollary 10.11 . Theorem 10.6 (iii) places the index corrections on the atom chain with every node quantity inherited, so Theorem 7.8 applies with $\mathsf a$ in place of $\nu$ . A spot correction leaves the atom chain and the clock alone by Theorem 10.9, hence every index row, and it changes the return of one step only, which enters the spot rows at that maturity and at later ones; earlier spot rows do not see it. Solving in the stated order therefore never disturbs a row already matched, and each stage is one of the two theorems applied to the chain in force.∎

References

[1] Erdinç Akyıldırım, Yan Dolinsky, and H. Mete Soner. Approximating stochastic volatility by recombinant trees. The Annals of Applied Probability, 24(5):2176–2205, 2014. doi: 10.1214/13-AAP977.

[2] David J. Aldous. Weak convergence and the general theory of processes. Unpublished monograph, University of California, Berkeley, 1981.

[3] Leif B. G. Andersen. Simple and efficient simulation of the Heston stochastic volatility model. Journal of Computational Finance, 11(3):1–42, 2008. doi: 10.21314/JCF.2008.189.

[4] Marco Avellaneda, Robert Buff, Craig Friedman, Nicolas Grandechamp, Lukasz Kruk, and Joshua Newman. Weighted Monte Carlo: A new technique for calibrating asset-pricing models. International Journal of Theoretical and Applied Finance, 4(1):91–119, 2001. doi: 10.1142/S0219024901000882.

[5] Julio Backhoff-Veraguas, Daniel Bartl, Mathias Beiglböck, and Manu Eder. Adapted Wasserstein distances and stability in mathematical finance. Finance and Stochastics, 24(3): 601–632, 2020. doi: 10.1007/s00780-020-00426-3.

[6] Julio Backhoff-Veraguas, Daniel Bartl, Mathias Beiglböck, and Manu Eder. All adapted topologies are equal. Probability Theory and Related Fields, 178(3):1125–1172, 2020. doi: 10.1007/s00440-020-00993-8.

[7] Julio Backhoff-Veraguas, Sigrid Källblad, and Benjamin A. Robinson. Adapted Wasserstein distance between the laws of SDEs. Stochastic Processes and their Applications, 189:104689, 2025. doi: 10.1016/j.spa.2025.104689.

[8] Daniel Bartl, Mathias Beiglböck, Gudmund Pammer, Stefan Schrott, and Xin Zhang. The Wasserstein space of stochastic processes in continuous time, 2025. arXiv:2501.14135.

[9] Daniel Bartl, Mathias Beiglböck, and Gudmund Pammer. The Wasserstein space of stochastic processes. Journal of the European Mathematical Society, 28(1):393–454, 2026. arXiv:2104.14245.

[10] Jocelyne Bion-Nadal and Denis Talay. On a Wasserstein-type distance between solutions to stochastic differential equations. The Annals of Applied Probability, 29(3), 2019. doi: 10.1214/18-AAP1423.

[11] Florian Bourgey and Julien Guyon. Fast exact joint S&P 500/VIX smile calibration in discrete and continuous time. Risk, February 2024. Cutting Edge: Volatility modelling; extended version available as SSRN working paper 4315084.

[12] Pierre Brémaud and Marc Yor. Changes of filtrations and of probability measures. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 45:269–295, 1978.

[13] Damiano Brigo. The general mixture-diffusion SDE and its relationship with an uncertain-volatility option model with volatility-asset decorrelation, 2008. arXiv:0812.4052.

[14] Damiano Brigo and Fabio Mercurio. Lognormal-mixture dynamics and calibration to market volatility smiles. International Journal of Theoretical and Applied Finance, 5(4): 427–446, 2002. doi: 10.1142/S0219024902001511.

[15] Mark Broadie and Ashish Jain. The effect of jumps and discrete sampling on volatility and variance swaps. International Journal of Theoretical and Applied Finance, 11(8):761–797, 2008. doi: 10.1142/S0219024908005032.

[16] Mark Broadie and Özgür Kaya. Exact simulation of stochastic volatility and other affine jump diffusion processes. Operations Research, 54(2):217–231, 2006. doi: 10.1287/opre.1050. 0247.

[17] Hans Buehler and Evgeny Ryskin. Discrete local volatility for large time steps, 2016. SSRN working paper 2783409.

[18] John Buffington and Robert J. Elliott. American options with regime switching. International Journal of Theoretical and Applied Finance, 5(5):497–514, 2002. doi: 10.1142/ S0219024902001523.

[19] François Coquet, Jean Mémin, and Leszek Słomiński. On weak convergence of filtrations. In Séminaire de Probabilités XXXV, volume 1755 of Lecture Notes in Mathematics, pages 306–328. Springer, 2001.

[20] Christa Cuchiero, Guido Gazzani, Janka Möller, and Sara Svaluto-Ferro. Joint calibration to SPX and VIX options with signature-based models. Mathematical Finance, 35(1):161–213, 2025. doi: 10.1111/mafi.12442.

[21] Zhenyu Cui, J. Lars Kirkby, and Duy Nguyen. A general valuation framework for SABR and stochastic local volatility models. SIAM Journal on Financial Mathematics, 9(2):520–563, 2018. doi: 10.1137/16M1106572.

[22] Zhenyu Cui, J. Lars Kirkby, and Duy Nguyen. Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations. European Journal of Operational Research, 290(3):1046–1062, 2021. doi: 10.1016/j.ejor.2020.09.008.

[23] Kresimir Demeterfi, Emanuel Derman, Michael Kamal, and Joseph Zou. More than you ever wanted to know about volatility swaps. Goldman Sachs Quantitative Strategies Research Notes, 1999.

[24] Emanuel Derman and Iraj Kani. Riding on a smile. Risk, 7(2):32–39, 1994.

[25] Ivan Guo, Grégoire Loeper, Jan Obłój, and Shiyi Wang. Joint modelling and calibration of SPX and VIX by optimal transport. SIAM Journal on Financial Mathematics, 13(1):1–31, 2022. doi: 10.1137/20M1375905.

[26] Julien Guyon. Dispersion-constrained martingale Schrödinger problems and the exact joint S&P 500/VIX smile calibration puzzle. Finance and Stochastics, 28:27–79, 2024. doi: 10.1007/s00780-023-00524-y.

[27] Martin Hairer and Jonathan C. Mattingly. Yet another look at Harris’ ergodic theorem for Markov chains. In Seminar on Stochastic Analysis, Random Fields and Applications VI, volume 63 of Progress in Probability, pages 109–117. Springer, Basel, 2011. doi: 10.1007/ 978-3-0348-0021-1_7.

[28] Douglas N. Hoover. Convergence in distribution and Skorokhod convergence for the general theory of processes. Probability Theory and Related Fields, 89:239–259, 1991. doi: 10.1007/bf01198786.

[29] Shaosai Huang. Common-budget certification of rolling SPX/VIX markets: Finite witnesses, exact continuous realization, and the clock boundary, 2026. Working manuscript.

[30] Shaosai Huang. Linked physical and risk-neutral SPX/VIX laws: The identified set under a divergence-bounded pricing kernel, 2026. Working manuscript.

[31] Jean Jacod. Grossissement initial, hypothèse (H $'$ ) et théorème de Girsanov. In Thierry Jeulin and Marc Yor, editors, Grossissements de filtrations: exemples et applications, volume 1118 of Lecture Notes in Mathematics, pages 15–35. Springer, 1985. doi: 10.1007/BFb0075768.

[32] Harold J. Kushner and Paul Dupuis. Numerical Methods for Stochastic Control Problems in Continuous Time. Springer, New York, 2nd edition, 2001.

[33] Lingfei Li and Gongqiu Zhang. Error analysis of finite difference and Markov chain approximations for option pricing. Mathematical Finance, 28(3):877–919, 2018. doi: 10.1111/ mafi.12161.

[34] Chia Chun Lo and Konstantinos Skindilias. An improved Markov chain approximation methodology: Derivatives pricing and model calibration. International Journal of Theoretical and Applied Finance, 17(7):1450047, 2014. doi: 10.1142/S0219024914500472.

[35] Aleksandar Mijatović and Martijn Pistorius. Continuously monitored barrier options under Markov processes. Mathematical Finance, 23(1):1–38, 2013. doi: 10.1111/j.1467-9965.2011. 00486.x.

[36] Vasanttilak Naik. Option valuation and hedging strategies with jumps in the volatility of asset returns. The Journal of Finance, 48(5):1969–1984, 1993. doi: 10.1111/j.1540-6261.1993. tb05137.x.

[37] Daniel B. Nelson and Krishna Ramaswamy. Simple binomial processes as diffusion approximations in financial models. Review of Financial Studies, 3(3):393–430, 1990. doi: 10.1093/rfs/3.3.393.

[38] Anthony Neuberger. The log contract. The Journal of Portfolio Management, 20(2):74–80, 1994. doi: 10.3905/jpm.1994.409478.

[39] Yann Ollivier. Ricci curvature of Markov chains on metric spaces. Journal of Functional Analysis, 256(3):810–864, 2009. doi: 10.1016/j.jfa.2008.11.001.

[40] James M. Ortega. The Newton–Kantorovich theorem. The American Mathematical Monthly, 75(6):658–660, 1968. doi: 10.2307/2313800.

[41] Georg Ch. Pflug and Alois Pichler. A distance for multistage stochastic optimization models. SIAM Journal on Optimization, 22(1):1–23, 2012. doi: 10.1137/110825054.

[42] Mark Rubinstein. Implied binomial trees. The Journal of Finance, 49(3):771–818, 1994. doi: 10.1111/j.1540-6261.1994.tb00079.x.

[43] Daniel Rudolf and Nikolaus Schweizer. Perturbation theory for Markov chains via Wasserstein distance. Bernoulli, 24(4A):2610–2639, 2018. doi: 10.3150/17-BEJ938.

[44] Gongqiu Zhang and Lingfei Li. Analysis of Markov chain approximation for option pricing and hedging: Grid design and convergence behavior. Operations Research, 2019. doi: 10.1287/opre.2018.1791.

How to cite

Shaosai Huang (2026). Certified Finite Schemes for Linked SPX/VIX Markets. Working paper, version of September 2026. Kspectra Research. SSRN 7435365 (doi:10.2139/ssrn.7435365). https://kspectra.ai/papers/certified-finite-schemes-spx-vix/

@misc{huang2026certified,
  author = {Huang, Shaosai},
  title  = {{Certified Finite Schemes for Linked SPX/VIX Markets}},
  year   = {2026},
  month  = sep,
  note   = {Working paper, version of September 2026},
  doi    = {10.2139/ssrn.7435365},
  url    = {https://kspectra.ai/papers/certified-finite-schemes-spx-vix/}
}

For AI tools and text processing, the full paper is also available as Markdown with LaTeX formulas.

$m$	$N_m$	$d^{\varphi }$	$d^{\mathsf m}$	$d^{\mathsf a}$	$d^{\sqrt {\mathsf s}}$	$U$ , $L^2$	$X_H$ , $L^2$	physical (s.e.)
0	108	$0$	$2.9\cdot 10^{-1}$	$6.2\cdot 10^{-2}$	$0$	$5.8\cdot 10^{-3}$	$2.2\cdot 10^{-2}$	$2.0\cdot 10^{-3}$ ( $1.2\cdot 10^{-3}$ )
1	338	$0$	$1.9\cdot 10^{-1}$	$4.4\cdot 10^{-2}$	$0$	$3.1\cdot 10^{-3}$	$1.3\cdot 10^{-2}$	$2.5\cdot 10^{-3}$ ( $9.3\cdot 10^{-4}$ )
2	1053	$0$	$1.1\cdot 10^{-1}$	$2.6\cdot 10^{-2}$	$0$	$1.6\cdot 10^{-3}$	$8.6\cdot 10^{-3}$	$1.2\cdot 10^{-3}$ ( $7.1\cdot 10^{-4}$ )