Second-order numerical schemes for decoupled forward-backward stochastic differential equations with jumps

TL;DR

This paper introduces second-order accurate numerical schemes for decoupled forward-backward stochastic differential equations with jumps, achieving high accuracy without jump-adapted discretization, validated by numerical examples.

Contribution

The paper develops a semi-discrete and a fully discrete second-order scheme for FBSDEs with jumps, including novel schemes for backward SDEs and quadrature-based spatial discretization.

Findings

Semi-discrete scheme achieves second-order convergence.

Fully discrete scheme with quadrature rules is highly accurate.

Numerical examples confirm effectiveness and precision.

Abstract

We propose new numerical schemes for decoupled forward-backward stochastic differential equations (FBSDEs) with jumps, where the stochastic dynamics are driven by a $d$-dimensional Brownian motion and an independent compensated Poisson random measure. A semi-discrete scheme is developed for discrete time approximation, which is constituted by a classic scheme for the forward SDE [17, 25] and a novel scheme for the backward SDE. Under some reasonable regularity conditions, we prove that the semi-discrete scheme can achieve second-order convergence in approximating the FBSDEs of interest; and such convergence rate does not require jump-adapted temporal discretization. Next, to add in spatial discretization, a fully discrete scheme is developed by designing accurate quadrature rules for estimating the involved conditional mathematical expectations. Several numerical examples are given to illustrate the effectiveness and the high accuracy of the proposed schemes.