A Monte Carlo method for backward stochastic differential equations with Hermite martingales

Abstract

Backward stochastic differential equations (BSDEs) appear in many problems in stochastic optimal control theory, mathematical finance, insurance and economics. This work deals with the numerical approximation of the class of Markovian BSDEs where the terminal condition is a functional of a Brownian motion. Using Hermite martingales, we show that the problem of solving a BSDE is identical to solving a countable infinite-dimensional system of ordinary differential equations (ODEs). The family of ODEs belongs to the class of stiff ODEs, where the associated functional is one-sided Lipschitz. On this basis, we derive a numerical scheme and provide numerical applications.

Numerical methods for backward stochastic differential equations: A survey

“…Some backward Euler type methods however do require some regularity assumptions, in particular smoothness and boundedness conditions [22,45,93] to ensure the theoretical convergence results hold, which can, for example, rely on the continuity and differentiability of the terminal condition. Least-squares regression based methods in general require a few extra regularity assumptions on the coefficients of the BSDE (and SDE) [17,51,79,89], so as to allow the derivation of robust estimates for the involved error using various regression tools and also to ensure the stability of the algorithm. Malliavin calculus based methods tend to require lots of extra assumptions, including various regularity assumptions and in particular, assumptions regarding the Malliavin weights [52,53,75].…”

“…The SGBM algorithm involves the approximation of conditional expectations by means of bundling Monte Carlo sample paths and a local regress-later technique within each bundle. By employing Hermite martingales, the authors of [79] formulate the problem of solving a FBSDE as the problem of solving a countably infinite-dimensional systems of ODEs. On this basis, they develop a numerical scheme which involves the projection of the solution onto generalized Hermite polynomials.…”