Convergence of particles and tree based scheme for singular FBSDEs

TL;DR

This paper presents a convergent particle and tree-based numerical scheme for solving singular forward-backward stochastic differential equations (FBSDEs) and related PDEs, validated through theoretical analysis and numerical experiments.

Contribution

It introduces a practical implementation of a splitting scheme for singular FBSDEs using particle and tree approximations, with proven convergence rates.

Findings

The numerical method converges under certain conditions.

The scheme effectively approximates solutions to singular FBSDEs.

Numerical experiments demonstrate the method's efficiency.

Abstract

We study an implementation of the theoretical splitting scheme introduced in [Chassagneux and Yang, 2022] for singular FBSDEs [Carmona and Delarue 2013] and their associated quasi-linear degenerate PDEs. The fully implementable algorithm is based on particles approximation of the transport operator and tree like approximation of the diffusion operator appearing in the theoretical splitting. We prove the convergence with a rate of our numerical method under some reasonable conditions on the coefficients functions. This validates a posteriori some numerical results obtained in [Chassagneux and Yang, 2022]. We conclude the paper with a numerical section presenting various implementations of the algorithm and discussing their efficiency in practice.