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On the orthogonal expansion of iterated Stratonovich stochastic integrals

Konstantin A. Rybakov

TL;DR

The paper addresses when orthogonal expansions of iterated Stratonovich stochastic integrals admit equal matrix traces and integral traces for a broad class of $L_2$ kernels. It develops a trace-class operator framework with an averaging diagonal-trace definition, proves a key diagonal-trace identity for $f(t,\tau)=\varphi(t)\psi(\tau)\,1(t-\tau)$ that extends previous smoothness requirements, and shows how to construct and decompose kernels to ensure trace consistency. The main result establishes that for $\varphi,\psi \in L_2(\mathds{T})$ and any basis of $L_2(\mathds{T})$, the trace of expansion coefficients matches the integral trace, with a clear pathway to generalizing to $f$ of the form $g+g^*$ and to $k$-fold cases via the $L_2^{\text{tr}(j_1\dots j_k)}(\mathds{T}^k)$ class. This provides a rigorous foundation for representing high-order Stratonovich integrals through orthogonal expansions, aiding numerical solutions of high-order stochastic differential equations.

Abstract

We consider a class of functions for which the multiple Stratonovich stochastic integral or equivalent iterated Stratonovich stochastic integral with square integrable weights is defined by the orthogonal expansion. The equality of the trace of expansion coefficients matrix for these functions and the corresponding integral trace is established.

On the orthogonal expansion of iterated Stratonovich stochastic integrals

TL;DR

The paper addresses when orthogonal expansions of iterated Stratonovich stochastic integrals admit equal matrix traces and integral traces for a broad class of kernels. It develops a trace-class operator framework with an averaging diagonal-trace definition, proves a key diagonal-trace identity for that extends previous smoothness requirements, and shows how to construct and decompose kernels to ensure trace consistency. The main result establishes that for and any basis of , the trace of expansion coefficients matches the integral trace, with a clear pathway to generalizing to of the form and to -fold cases via the class. This provides a rigorous foundation for representing high-order Stratonovich integrals through orthogonal expansions, aiding numerical solutions of high-order stochastic differential equations.

Abstract

We consider a class of functions for which the multiple Stratonovich stochastic integral or equivalent iterated Stratonovich stochastic integral with square integrable weights is defined by the orthogonal expansion. The equality of the trace of expansion coefficients matrix for these functions and the corresponding integral trace is established.

Paper Structure

This paper contains 3 sections, 4 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Necessary definitions and preliminary results
  3. The main result

Key Result

Theorem 1

Let the linear operator $\operatorname{F} \colon L_2(\mathds{T}) \to L_2(\mathds{T})$ with the kernel $f \in L_2(\mathds{T}^2)$ is a trace class operator, and $\{q_i\}$ is a basis of $L_2(\mathds{T})$. Then where $F_{ij}$ are expansion coefficients eq1 of the function $f$ relative to the basis $\{q_i q_j\}$, and the integral on the right-hand side is understood as follows: i.e., $\operatorname{S

Theorems & Definitions (6)

  • Theorem 1: see Ref5Ref6
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Proposition 2