Deep Operator BSDE: a Numerical Scheme to Approximate Solution Operators

TL;DR

This paper introduces a neural network-based numerical scheme for approximating solution operators of BSDEs, combining Wiener chaos decomposition and Euler methods, with proven convergence and practical numerical demonstrations.

Contribution

It presents a novel numerical approach integrating Wiener chaos and neural networks to efficiently approximate BSDE solution operators with convergence guarantees.

Findings

The scheme converges under mild assumptions.

The method achieves accurate approximations in numerical examples.

Convergence rate established in restrictive cases.

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.