Numerical simulation of quadratic BSDEs

TL;DR

This paper develops a numerical method for solving quadratic Markovian BSDEs, providing stability analysis, error estimates, and a practical algorithm with numerical demonstrations.

Contribution

It introduces a new discretization scheme for quadratic BSDEs, along with stability, error bounds, and an implementable quantization-based algorithm.

Findings

Established a comparison theorem and stability results for the scheme.

Derived a discretization error bound of order 1/2 minus epsilon.

Provided numerical examples demonstrating the convergence of the method.

Abstract

This article deals with the numerical approximation of Markovian backward stochastic differential equations (BSDEs) with generators of quadratic growth with respect to $z$ and bounded terminal conditions. We first study a slight modification of the classical dynamic programming equation arising from the time-discretization of BSDEs. By using a linearization argument and BMO martingales tools, we obtain a comparison theorem, a priori estimates and stability results for the solution of this scheme. Then we provide a control on the time-discretization error of order $\frac{1}{2}-\varepsilon$ for all $\varepsilon>0$. In the last part, we give a fully implementable algorithm for quadratic BSDEs based on quantization and illustrate our convergence results with numerical examples.