Fully Discrete Schemes and Their Analyses for Forward-Backward Stochastic Differential Equations

TL;DR

This paper introduces new fully discrete numerical schemes for solving forward-backward stochastic differential equations (FBSDEs) using transposition solutions, time-splitting, and transition semigroups, with proven convergence and demonstrated effectiveness in financial applications.

Contribution

It presents novel numerical schemes for FBSDEs based on transposition solutions, combining time-splitting and discrete transition semigroups, with convergence analysis and practical financial applications.

Findings

Schemes converge for Lipschitz and monotone drivers.

Numerical experiments show effectiveness in financial derivatives.

Methods can be extended to high-order schemes for FBSDEs.

Abstract

We propose some numerical schemes for forward-backward stochastic differential equations (FBSDEs) based on a new fundamental concept of transposition solutions. These schemes exploit time-splitting methods for the variation of constants formula of the associated partial differential equations and a discrete representation of the transition semigroups. The convergence of the schemes is established for FBSDEs with uniformly Lipschitz drivers, locally Lipschitz and maximal monotone drivers. Numerical experiments are presented for several nonlinear financial derivative pricing problems to demonstrate the adaptivity and effectiveness of the new schemes. The ideas here can be applied to construct high-order schemes for FBSDEs with general Markov forward processes.