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

TL;DR

This paper develops Malliavin calculus techniques for backward stochastic differential equations (BSDEs) with general terminal conditions and generators, providing new regularity results and convergence rates for numerical schemes.

Contribution

It introduces a Malliavin calculus framework for BSDEs with non-standard terminal conditions and generators, enabling new regularity and numerical analysis results.

Findings

Established $L^p$-Hölder continuity of BSDE solutions.

Constructed numerical schemes with proven convergence rates.

Extended Malliavin calculus applications to general BSDEs.

Abstract

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.