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arxiv: 2510.06981 · v4 · submitted 2025-10-08 · 🧮 math.PR

New representations of the Hu-Meyer formulas and series expansion of iterated Stratonovich stochastic integrals with respect to components of a multidimensional Wiener process

Dmitriy F. Kuznetsov This is my paper

Pith reviewed 2026-05-18 09:13 UTC · model grok-4.3

classification 🧮 math.PR MSC 60H0560H1060H35
keywords Hu-Meyer formulasStratonovich integralsWiener processmultiple stochastic integralsFourier seriesnumerical methods for SDEs
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The pith

Hu-Meyer formulas receive new representations that convert multiple Wiener integrals into sums of Stratonovich integrals for multidimensional Wiener processes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives explicit formulas that express a multiple Wiener stochastic integral as a sum of multiple Stratonovich stochastic integrals and the reverse conversion for a multidimensional Wiener process. It applies several definitions of the multiple Stratonovich integral together with different sufficient conditions, including those based on generalized multiple Fourier series. The work extends earlier verifications of those Fourier-series conditions from low-multiplicity cases with arbitrary bases and special bases at multiplicities 7-8 to the full multidimensional setting. A sympathetic reader would care because these conversions support the construction of high-order strong numerical schemes for systems of Itô stochastic differential equations that do not commute.

Core claim

The formulas expressing a multiple Wiener stochastic integral through the sum of multiple Stratonovich stochastic integrals and the formula expressing a multiple Stratonovich stochastic integral through the sum of multiple Wiener stochastic integrals are derived for the case of a multidimensional Wiener process using several different definitions of the multiple Stratonovich stochastic integral and several variants of sufficient conditions for the validity of the Hu-Meyer formulas; in particular the 2006 proof method based on generalized multiple Fourier series is applied and the corresponding sufficient conditions are verified for iterated Stratonovich integrals with respect to components.

What carries the argument

Hu-Meyer formulas based on generalized multiple Fourier series, which convert between multiple Wiener and multiple Stratonovich integrals under stated integrability conditions.

If this is right

  • High-order strong numerical methods for non-commutative Itô SDEs can be built directly from the converted integral expansions.
  • Series expansions of iterated Stratonovich integrals become available for simulation in any finite dimension.
  • The same conversion formulas apply under each of the listed definitions of the Stratonovich integral.
  • Numerical schemes no longer need separate handling for the commutative and non-commutative cases once the formulas are implemented.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Fourier-series technique could be tested on other semimartingales whose quadratic variation is absolutely continuous.
  • If the conditions hold in every finite dimension they may also hold for cylindrical Wiener processes in Hilbert space.
  • Implementation in existing SDE solvers would reduce the number of random variables that must be simulated at each step.

Load-bearing premise

The sufficient conditions based on generalized multiple Fourier series continue to hold when the underlying process is a multidimensional Wiener process rather than a one-dimensional one.

What would settle it

An explicit computation for multiplicity 9 or higher with a non-special basis in which the Fourier-series remainder term fails to vanish would show that the verified conditions do not extend.

read the original abstract

The article is devoted to the systematic derivation of new representations of the Hu-Meyer formulas. The formula expressing a multiple Wiener stochastic integral through the sum of multiple Stratonovich stochastic integrals and the formula expressing a multiple Stratonovich stochastic integral through the sum of multiple Wiener stochastic integrals are derived for the case of a multidimensional Wiener process. At that several different definitions of the multiple Stratonovich stochastic integral and several variants of sufficient conditions for the validity of the Hu-Meyer formulas are used. In particular, the proof method proposed by the author in 2006 is applied to obtain Hu-Meyers formulas based on generalized multiple Fourier series for the case of a multidimensional Wiener process. Of great importance for the numerical solution of Ito stochastic differential equations is the verification of sufficient conditions for the applicability of the Hu-Meyer formula (based on generalized multiple Fourier series) for the case of iterated Stratonovich stochastic integrals with respect to components of a multidimensional Wiener process. In the author's previous works, the indicated conditions were verified for iterated Stratonovich stochastic integrals of multiplicities 1 to 6 (the case of an arbitrary basis in the Hilbert space) and for iterated Stratonovich stochastic integrals of multiplicities 7 and 8 (the case of two special bases in Hilbert space (the trigonometric Fourier basis and the basis of Legendre polynomials)). Therefore, the results of the article will be usefull for constructing high-order strong numerical methods for non-commutative systems of Ito stochastic differential equations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript derives new representations of the Hu-Meyer formulas expressing multiple Wiener stochastic integrals in terms of multiple Stratonovich integrals (and conversely) for a multidimensional Wiener process. It employs several definitions of the multiple Stratonovich integral together with multiple variants of sufficient conditions, including the author's 2006 generalized multiple Fourier series method. The central contribution is the verification that these sufficient conditions hold for iterated Stratonovich integrals with respect to components of the multidimensional Wiener process, extending the author's prior verifications (multiplicities 1-6 for arbitrary bases and 7-8 for trigonometric and Legendre bases).

Significance. If the derivations and verifications are correct, the results would directly support construction of high-order strong numerical methods for non-commutative systems of Itô SDEs by supplying the required Hu-Meyer conversion formulas in the multidimensional setting. The systematic use of several definitions and condition variants adds robustness; the explicit extension of the 2006 Fourier-series technique to the multidimensional case is a clear technical advance over the author's earlier one-dimensional or low-multiplicity results.

major comments (2)
  1. [Verification of sufficient conditions] Verification section (following the 2006 method): the claim that the generalized multiple Fourier series sufficient conditions remain valid for mixed-component iterated Stratonovich integrals (e.g., those involving distinct components W^i and W^j with i≠j) is not supported by a separate convergence or bound argument. Because d⟨W^i,W^j⟩=δ_{ij}dt, the quadratic-variation correction terms are absent for i≠j; the manuscript must explicitly confirm that the Fourier-series remainder estimates still hold under these reduced corrections, as this is load-bearing for applicability to non-commutative SDEs.
  2. [Application of 2006 method] Section applying the 2006 proof method to the multidimensional case: the argument reduces the new verification to the author's prior results for multiplicities 1-6 (arbitrary basis) and 7-8 (special bases) plus standard Fourier techniques, but does not supply an explicit check or counter-example for index patterns where component indices differ. Without this, the extension to the full multidimensional setting rests on an unverified inductive step.
minor comments (2)
  1. [Abstract] Abstract, line near end: 'usefull' is a typographical error and should read 'useful'.
  2. [Abstract] Abstract, opening sentence: 'At that several different' is awkward; consider 'Using several different' or 'Employing several different'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. The points raised regarding the verification of sufficient conditions for mixed-component integrals and the explicit application of the 2006 method in the multidimensional setting are well taken. We address each major comment below and have revised the manuscript to incorporate additional clarifications and an illustrative example.

read point-by-point responses

  1. Referee: Verification section (following the 2006 method): the claim that the generalized multiple Fourier series sufficient conditions remain valid for mixed-component iterated Stratonovich integrals (e.g., those involving distinct components W^i and W^j with i≠j) is not supported by a separate convergence or bound argument. Because d⟨W^i,W^j⟩=δ_{ij}dt, the quadratic-variation correction terms are absent for i≠j; the manuscript must explicitly confirm that the Fourier-series remainder estimates still hold under these reduced corrections, as this is load-bearing for applicability to non-commutative SDEs.

    Authors: We appreciate the referee's observation on the distinction for cross terms. When i ≠ j, the iterated Stratonovich integral with respect to distinct Wiener components does not involve quadratic covariation corrections, making it equivalent to the iterated Itô integral. The sufficient conditions from the 2006 generalized multiple Fourier series method rely on L²-convergence of the series expansions and the decay of Fourier coefficients under square-integrability assumptions on the kernels. These estimates are derived from the orthonormal basis properties in the Hilbert space and hold uniformly, independent of the specific quadratic (co)variation structure of the driving processes. The absence of correction terms for i ≠ j does not alter the remainder bounds. We have added an explicit confirming statement in the verification section of the revised manuscript. revision: yes

  2. Referee: Section applying the 2006 proof method to the multidimensional case: the argument reduces the new verification to the author's prior results for multiplicities 1-6 (arbitrary basis) and 7-8 (special bases) plus standard Fourier techniques, but does not supply an explicit check or counter-example for index patterns where component indices differ. Without this, the extension to the full multidimensional setting rests on an unverified inductive step.

    Authors: The reduction is justified because the 2006 method is formulated in a general Hilbert-space setting, and the components of the multidimensional Wiener process correspond to orthogonal directions. Prior verifications for arbitrary bases at multiplicities 1–6 therefore extend directly to selections of distinct components via the tensor-product structure. Nevertheless, we agree that an explicit check for a mixed-index pattern strengthens the presentation. We have inserted a representative example with differing component indices in the revised section to illustrate the inductive step. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations extend prior method independently

full rationale

The paper derives new Hu-Meyer representations for the multidimensional Wiener process by applying the author's 2006 Fourier-series method and verifies the sufficient conditions directly for iterated Stratonovich integrals with respect to its components. Prior author works are cited only for the lower-multiplicity cases (1-6 arbitrary, 7-8 special bases), while the multidimensional extension and mixed-component applicability are presented as the current contribution with its own proofs and conditions. No step reduces a claimed result to a fitted input, self-definition, or unverified self-citation chain; the central claims remain independent mathematical derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only: the derivations rely on standard properties of multiple stochastic integrals and Fourier series expansions in Hilbert space; no free parameters or new entities are mentioned.

axioms (1)
  • domain assumption Standard definitions and conversion rules for multiple Wiener and Stratonovich integrals hold under the chosen variants of sufficient conditions.
    Abstract invokes several different definitions and variants of sufficient conditions for Hu-Meyer formulas without deriving them from more primitive axioms.

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    Relation between the paper passage and the cited Recognition theorem.

    The formula expressing a multiple Wiener stochastic integral through the sum of multiple Stratonovich stochastic integrals ... for the case of a multidimensional Wiener process ... verification of sufficient conditions ... based on generalized multiple Fourier series

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Works this paper leans on

42 extracted references · 42 canonical work pages · 1 internal anchor

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