Weak approximation of martingale representations

TL;DR

This paper introduces a systematic method for explicitly approximating martingale representations for a broad class of Brownian functionals, without requiring Markovian or differentiability assumptions, and provides convergence rates.

Contribution

It develops a novel approach to approximate martingale integrands using directional derivatives of the weak Euler scheme, applicable to path-dependent SDEs.

Findings

Explicit convergence rates for Lipschitz functionals

Method works without Markov property or differentiability

Provides consistent estimators for martingale integrands

Abstract

We present a systematic method for computing explicit approximations to martingale representations for a large class of Brownian functionals. The approximations are obtained by obtained by computing a directional derivative of the weak Euler scheme and yield a consistent estimator for the integrand in the martingale representation formula for any square-integrable functional of the solution of an SDE with path-dependent coefficients. Explicit convergence rates are derived for functionals which are Lipschitz-continuous in the supremum norm. Our results require neither the Markov property, nor any differentiability conditions on the functional or the coefficients of the stochastic differential equations involved.