Numerical Methods and Analysis via Random Field Based Malliavin Calculus for Backward Stochastic PDEs

TL;DR

This paper develops numerical methods and theoretical analysis for a class of complex backward stochastic PDEs with nonlinear and high-order operators, utilizing novel Malliavin calculus techniques for existence, uniqueness, and convergence.

Contribution

It introduces a unified approach combining random field Malliavin calculus with discrete numerical schemes for B-SPDEs, addressing existence, uniqueness, and convergence under random environments.

Findings

Established existence and uniqueness of solutions under generalized conditions

Developed discrete numerical schemes with proven convergence rates

Extended Malliavin calculus to handle high-order and nonlinear operators

Abstract

We study the adapted solution, numerical methods, and related convergence analysis for a unified backward stochastic partial differential equation (B-SPDE). The equation is vector-valued, whose drift and diffusion coefficients may involve nonlinear and high-order partial differential operators. Under certain generalized Lipschitz and linear growth conditions, the existence and uniqueness of adapted solution to the B-SPDE are justified. The methods are based on completely discrete schemes in terms of both time and space. The analysis concerning error estimation or rate of convergence of the methods is conducted. The key of the analysis is to develop new theory for random field based Malliavin calculus to prove the existence and uniqueness of adapted solutions to the first-order and second-order Malliavin derivative based B-SPDEs under random environments.