Malliavin calculus for backward stochastic differential equations and application to numerical solutions

Yaozhong Hu, David Nualart, Xiaoming Song

Published 2012-02-21Version 1

In this paper we study backward stochastic differential equations with general terminal value and general random generator. In particular, we do not require the terminal value be given by a forward diffusion equation. The randomness of the generator does not need to be from a forward equation, either. Motivated from applications to numerical simulations, first we obtain the $L^p$-H\"{o}lder continuity of the solution. Then we construct several numerical approximation schemes for backward stochastic differential equations and obtain the rate of convergence of the schemes based on the obtained $L^p$-H\"{o}lder continuity results. The main tool is the Malliavin calculus.