Time discretization of FBSDE with polynomial growth drivers and reaction-diffusion PDEs

TL;DR

This paper analyzes the error and stability of time discretization schemes for FBSDEs with polynomial growth drivers, demonstrating convergence of a tamed explicit Euler scheme and extending regularity results.

Contribution

It provides the first detailed stability and convergence analysis for explicit schemes applied to FBSDEs with polynomial growth drivers, including a tamed Euler scheme.

Findings

Explicit Euler scheme may diverge for polynomial growth drivers.

A tamed explicit Euler scheme converges under certain conditions.

Extended regularity results for FBSDEs with polynomial growth drivers.

Abstract

In this paper, we undertake the error analysis of the time discretization of systems of Forward-Backward Stochastic Differential Equations (FBSDEs) with drivers having polynomial growth and that are also monotone in the state variable. We show with a counter-example that the natural explicit Euler scheme may diverge, unlike in the canonical Lipschitz driver case. This is due to the lack of a certain stability property of the Euler scheme which is essential to obtain convergence. However, a thorough analysis of the family of $\theta$-schemes reveals that this required stability property can be recovered if the scheme is sufficiently implicit. As a by-product of our analysis, we shed some light on higher order approximation schemes for FBSDEs under non-Lipschitz condition. We then return to fully explicit schemes and show that an appropriately tamed version of the explicit Euler scheme enjoys the required stability property and as a consequence converges. In order to establish convergence of the several discretizations, we extend the canonical path- and first-order variational regularity results to FBSDEs with polynomial growth drivers which are also monotone. These results are of independent interest for the theory of FBSDEs.