Full-Projection explicit FBSDE scheme for parabolic PDEs with superlinear nonlinearities

TL;DR

This paper introduces a stable, explicit time-discretization scheme for high-dimensional parabolic PDEs with superlinear nonlinearities, leveraging FBSDEs and quantization methods for improved approximation accuracy.

Contribution

It develops a novel Full-Projection explicit scheme for FBSDEs with polynomial growth drivers, ensuring stability and convergence in high-dimensional settings.

Findings

The scheme preserves stability properties of the continuous dynamics.

Convergence rates are established for the proposed discretization.

Numerical examples demonstrate the scheme's effectiveness.

Abstract

Developing efficient and stable approximations for high dimensional PDEs is of key importance for numerous applications. The language of Forward-Backward Stochastic Differential Equations (FBSDE), with its nonlinear Feynman-Kac formula, allows for purely probabilistic representations of the solution and its gradient for parabolic nonlinear PDEs. In this work we build on the recent results of [Lionnet, dos Reis and Szpruch 2015] by introducing and studying a Full-Projection explicit time-discretization scheme for the approximation of FBSDEs with non-globally Lipschitz drivers of polynomial growth. We establish convergence rates and we show that, unlike classical explicit schemes, it preserves stability properties present in the continuous-time dynamics, in particular, the scheme is able to preserve the possible coercivity/contraction property of the PDE's coefficients. The scheme is then coupled with a quantization-type approximation of the conditional expectations on a space-time grid in order to provide a complete approximation scheme for these FBSDEs/nonlinear PDEs and a full analysis is also carried out. We illustrate our findings with numerical examples.