ODE-Based Multistep Schemes for Backward Stochastic Differential Equations

In this paper, we explore a new approach to design and analyze numerical schemes for backward stochastic differential equations (BSDEs). By the nonlinear Feynman-Kac formula, we reformulate the BSDE into a pair of reference ordinary differential equations (ODEs), which can be directly discretized by many standard ODE solvers, yielding the corresponding numerical schemes for BSDEs. In particular, by applying strong stability preserving (SSP) time discretizations to the reference ODEs, we can propose new SSP multistep schemes for BSDEs. Theoretical analyses are rigorously performed to prove the consistency, stability and convergency of the proposed SSP multistep schemes. Numerical experiments are further carried out to verify our theoretical results and the capacity of the proposed SSP multistep schemes for solving complex associated problems.

Automated summary

In this paper, a new approach for designing and analyzing numerical schemes for backward stochastic differential equations (BSDEs) is explored. The BSDEs are reformulated into a pair of reference ordinary differential equations (ODEs) using the nonlinear Feynman-Kac formula, which can be discretized directly by standard ODE solvers, resulting in corresponding numerical schemes. Furthermore, new strong stability preserving (SSP) multistep schemes for BSDEs are proposed by applying SSP time discretizations to the reference ODEs. Theoretical analyses and numerical experiments are performed to demonstrate the consistency, stability, and convergence of the proposed SSP multistep schemes.