Annals of Applied Probability

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

Arnaud Lionnet, Goncalo dos Reis, and Lukasz Szpruch

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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.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 4 (2018), 2544-2591.

Dates
Received: July 2016
Revised: July 2017
First available in Project Euclid: 9 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1533780280

Digital Object Identifier
doi:10.1214/17-AAP1366

Mathematical Reviews number (MathSciNet)
MR3843836

Zentralblatt MATH identifier
06974758

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

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

Citation

Lionnet, Arnaud; dos Reis, Goncalo; Szpruch, Lukasz. Convergence and qualitative properties of modified explicit schemes for BSDEs with polynomial growth. Ann. Appl. Probab. 28 (2018), no. 4, 2544--2591. doi:10.1214/17-AAP1366. https://projecteuclid.org/euclid.aoap/1533780280


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