A Chen-Fliess approximation for diffusion functionals

TL;DR

This paper introduces a Chen-Fliess series approximation for diffusion functionals of stochastic differential equations, extending stochastic Taylor expansions with error estimates and an intuitive interpretation via functional derivatives.

Contribution

It generalizes stochastic Taylor expansions by incorporating Chen-Fliess series and provides an L^{2} error estimate using functional derivatives.

Findings

Chen-Fliess series effectively approximates diffusion functionals

Provides L^{2} error bounds for the approximation

Connects coefficients to functional derivatives for intuitive understanding

Abstract

We show that an interesting class of functionals of stochastic differential equations can be approximated by a Chen-Fliess series of iterated stochastic integrals and give a L^{2} error estimate, thus generalizing the standard stochastic Taylor expansion. The coefficients in this series are given a very intuitive meaning by using functional derivatives, recently introduced by B. Dupire.