Convergence and qualitative properties of modified explicit schemes for BSDEs with polynomial growth

Research output: Contribution to journalArticlepeer-review

Authors

Colleges, School and Institutes

External organisations

  • University of Edinburgh, The

Abstract

The theory of Forward–Backward Stochastic Differential Equations (FBSDEs) paves a way to probabilistic numerical methods for nonlinear parabolic PDEs. The majority of the results on the numerical methods for FBSDEs relies on the global Lipschitz assumption, which is not satisfied for a number of important cases such as the Fisher–KPP or the FitzHugh–Nagumo equations. Furthermore, it has been shown in [Ann. Appl. Probab. 25 (2015) 2563–2625] that for BSDEs with monotone drivers having polynomial growth in the primary variable Y, only the (sufficiently) implicit schemes converge. But these require an additional computational effort compared to explicit schemes.

This article develops a general framework that allows the analysis, in a systematic fashion, of the integrability properties, convergence and qualitative properties (e.g., comparison theorem) for whole families of modified explicit schemes. The framework yields the convergence of some modified explicit scheme with the same rate as implicit schemes and with the computational cost of the standard explicit scheme.

To illustrate our theory, we present several classes of easily implementable modified explicit schemes that can computationally outperform the implicit one and preserve the qualitative properties of the solution to the BSDE. These classes fit into our developed framework and are tested in computational experiments.

Details

Original languageEnglish
Pages (from-to)2544-2591
Number of pages48
JournalAnnals of Applied Probability
Volume28
Issue number4
Publication statusPublished - 9 Aug 2018

Keywords

  • FBSDE, monotone driver, polynomial growth time, discretization, modified explicit schemes, nonexplosion, numerical stability