Deep Operator BSDE: a Numerical Scheme to Approximate Solution Operators

Abstract

Motivated by dynamic risk measures and conditional $g$-expectations, in this work we propose a numerical method to approximate the solution operator given by a Backward Stochastic Differential Equation (BSDE). The main ingredients for this are the Wiener chaos decomposition and the classical Euler scheme for BSDEs. We show convergence of this scheme under very mild assumptions, and provide a rate of convergence in more restrictive cases. We then implement it using neural networks, and we present several numerical examples where we can check the accuracy of the method.

References in corpus (10)

• DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
• Solving high-dimensional partial differential equations using deep learning
• A regression-based Monte Carlo method to solve backward stochastic differential equations
• Convergence of the Deep BSDE Method for Coupled FBSDEs
• Simulation of BSDEs by Wiener chaos expansion
• Malliavin calculus for backward stochastic differential equations and application to numerical solutions
• Approximation of backward stochastic differential equations using Malliavin weights and least-squares regression
• Numerical methods for backward stochastic differential equations: A survey
• Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing
• Unbiased deep solvers for linear parametric PDEs