Numerical approximation of general Lipschitz BSDEs with branching processes

TL;DR

This paper extends a branching process-based numerical algorithm for Lipschitz BSDEs to handle nonlinearities involving the solution's gradient, ensuring convergence and demonstrating effectiveness through simulations.

Contribution

It introduces a novel approach to approximate Lipschitz BSDEs with gradient-dependent nonlinearities using a localization and face-lifting procedure.

Findings

Algorithm converges without time horizon restrictions

Numerical simulations confirm the method's effectiveness

Handles gradient-dependent nonlinearities in BSDEs

Abstract

We extend the branching process based numerical algorithm of Bouchard et al. [3], that is dedicated to semilinear PDEs (or BSDEs) with Lipschitz nonlinearity, to the case where the nonlinearity involves the gradient of the solution. As in [3], this requires a localization procedure that uses a priori estimates on the true solution, so as to ensure the well-posedness of the involved Picard iteration scheme, and the global convergence of the algorithm. When, the nonlinearity depends on the gradient, the later needs to be controlled as well. This is done by using a face-lifting procedure. Convergence of our algorithm is proved without any limitation on the time horizon. We also provide numerical simulations to illustrate the performance of the algorithm.