The Annals of Applied Probability

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

Arnaud Lionnet, Gonçalo dos Reis, and Lukasz Szpruch

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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.

Article information

Ann. Appl. Probab., Volume 25, Number 5 (2015), 2563-2625.

Received: September 2013
Revised: June 2014
First available in Project Euclid: 30 July 2015

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Zentralblatt MATH identifier

Primary: 65C30: Stochastic differential and integral equations 60H35: Computational methods for stochastic equations [See also 65C30]
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H30: Applications of stochastic analysis (to PDE, etc.)

FBSDE monotone driver polynomial growth time discretization path regularity calculus of variations numerical schemes


Lionnet, Arnaud; dos Reis, Gonçalo; Szpruch, Lukasz. Time discretization of FBSDE with polynomial growth drivers and reaction–diffusion PDEs. Ann. Appl. Probab. 25 (2015), no. 5, 2563--2625. doi:10.1214/14-AAP1056.

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