Numerical approximation of singular Forward-Backward SDEs

TL;DR

This paper develops a splitting numerical scheme for singular fully coupled forward-backward stochastic differential equations with degenerate and non-smooth features, demonstrating convergence and effectiveness in high dimensions.

Contribution

It introduces a novel splitting approach combining neural networks and finite differences to approximate singular FBSDEs, addressing limitations of classical methods.

Findings

Convergence rate of 1/2 for the proposed scheme.

Effective in high-dimensional settings.

Numerical tests show very good results.

Abstract

In this work, we study the numerical approximation of a class of singular fully coupled forward backward stochastic differential equations. These equations have a degenerate forward component and non-smooth terminal condition. They are used, for example, in the modeling of carbon market[9] and are linked to scalar conservation law perturbed by a diffusion. Classical FBSDEs methods fail to capture the correct entropy solution to the associated quasi-linear PDE. We introduce a splitting approach that circumvent this difficulty by treating differently the numerical approximation of the diffusion part and the non-linear transport part. Under the structural condition guaranteeing the well-posedness of the singular FBSDEs [8], we show that the splitting method is convergent with a rate $1/2$. We implement the splitting scheme combining non-linear regression based on deep neural networks and conservative finite difference schemes. The numerical tests show very good results in possibly high dimensional framework.