Stein approximation for Itô and Skorohod integrals by Edgeworth type expansions

Nicolas Privault

doi:10.1214/ecp.v20-3827

Abstract

We derive Edgeworth-type expansions for Skorohod and Itô integrals with respect to Brownian motion, based on cumulant operators defined by the Malliavin calculus. As a consequence we obtain Stein approximation bounds for stochastic integrals, which apply to SDE solutions and to multiple stochastic integrals.

Highlights

Classical Edgeworth series around the Gaussian cumulative distribution function Φ(x) take the formΦ(x) + c1φ(x)H1(x) + · · · + cmφ(x)Hm(x) + · · ·, where φ(x), x ∈ R, is the standard Gaussian density, Hk(x) is the Hermite polynomial of degree k ≥ 1, and ck is a coefficient depending on the sequence of cumulantsn≥1 of a random variable F, cf. Chapter 5 of [6] and § A.4 of [11]
Edgeworth expansions are used in particular as asymptotic expansions for the cumulative distribution function P (F ≤ x) (or, in more general forms, as asymptotic expansions for expectations of the type E[h(F )], where h is some test function - see [6]), when F is centered with unit variance E[F 2] = 1, for example when F is a renormalized sum of independent random variables that can be approximated by the central limit theorem, cf
In the proposition we extend (2.1) into an expansion of all orders that will be applied to Stein approximation

Summary

Introduction

Classical Edgeworth series around the Gaussian cumulative distribution function Φ(x) take the form. Have been obtained by the Malliavin calculus in [8], [2], [3], written here for F a centered random variable with unit variance, where Γn+1 is a cumulant type operator on the Wiener space satisfying the relation n!E[ΓnF ] = κn+1, n ∈ N, cf [10]. This approach refines and extends the application of the Malliavin calculus to Stein approximation, Berry-Esseen bounds and the fourth moment theorem initiated in [9], see [12], and [7] for a review.

Edgeworth type expansions

Stein approximation