Numerical Methods for Backward Stochastic Differential Equations

Summary

Introduction

Backward stochastic differential equations (BSDEs) were introduced by Pardoux & Peng (1990) to give a probabilistic representation for the solutions of certain nonlinear partial differential equations, thus generalizing the Feynman- Kac formula.

This sort of equation has also found many applications in finance, notably in contingent claim valuation when there are constraints on the hedging portfolios (see El Karoui & Quenez 1995, El Karoui, Peng & Quenez 1994, Cvitanic & Karatzas 1992) and in the definition of stochastic differential utility (see Duffie & Epstein 1992, El Karoui, Peng & Quenez 1994). A financial application of forward-backward SDEs can be found in Duffie, Ma & Yong (1993).

However little research has yet been performed on numerical methods for BSDEs. Here we give a review of three different contributions in that field.

In Section 2, we present a random time discretization scheme introduced by V. Bally to approximate BSDEs. The advantage of Bally's scheme is that one can get a convergence result with virtually no other regularity assumption than the ones needed for the existence of a solution to the equation. However that scheme is not fully numerical and its actual implementation would require further approximations.

In Section 3, we give an account of a four step algorithm developed by J. Ma and P. Protter to solve a class of more general equations called forwardbackward SDEs. It is based on solving the associated PDE by a deterministictype method and also makes use of the Euler scheme for stochastic differential equations.