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

In this paper we undertake the error analysis of the time discretization of systems of ForwardBackward Stochastic Di erential 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 θ-schemes reveals that this required stability property can be recovered if the scheme is su ciently 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 pathand rst 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. 2010 AMS subject classi cations: Primary: 65C30; Secondary: 60H07, 60H30.