Mean square rate of convergence for random walk approximation of forward-backward SDEs

TL;DR

This paper analyzes the convergence rate of a random walk approximation for forward-backward stochastic differential equations, utilizing Malliavin calculus and PDE properties to provide explicit estimates based on smoothness.

Contribution

It introduces a new approach to estimate the L2 convergence rate of random walk approximations for FBSDEs using discretized Malliavin calculus and PDE analysis.

Findings

Established L2 convergence of (Y_n, Z_n) to (Y, Z)

Derived explicit convergence rates depending on smoothness of terminal condition

Utilized stochastic methods to analyze PDE and finite difference properties

Abstract

Let (Y, Z) denote the solution to a forward-backward SDE. If one constructs a random walk B n from the underlying Brownian motion B by Skorohod embedding, one can show L 2 convergence of the corresponding solutions (Y n , Z n) to (Y, Z). We estimate the rate of convergence in dependence of smoothness properties, especially for a terminal condition function in C 2,$\alpha$. The proof relies on an approximative representation of Z n and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the PDE associated to the FBSDE as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by stochastic methods.