The Annals of Applied Probability

Malliavin calculus for backward stochastic differential equations and application to numerical solutions

Yaozhong Hu, David Nualart, and Xiaoming Song

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In this paper we study backward stochastic differential equations with general terminal value and general random generator. In particular, we do not require the terminal value be given by a forward diffusion equation. The randomness of the generator does not need to be from a forward equation, either. Motivated from applications to numerical simulations, first we obtain the Lp-Hölder continuity of the solution. Then we construct several numerical approximation schemes for backward stochastic differential equations and obtain the rate of convergence of the schemes based on the obtained Lp-Hölder continuity results. The main tool is the Malliavin calculus.

Article information

Ann. Appl. Probab., Volume 21, Number 6 (2011), 2379-2423.

First available in Project Euclid: 23 November 2011

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Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05] 60H35: Computational methods for stochastic equations [See also 65C30] 65C30: Stochastic differential and integral equations 91G60: Numerical methods (including Monte Carlo methods)

Backward stochastic differential equations Malliavin calculus explicit scheme implicit scheme Clark–Ocone–Haussman formula rate of convergence Hölder continuity of the solutions


Hu, Yaozhong; Nualart, David; Song, Xiaoming. Malliavin calculus for backward stochastic differential equations and application to numerical solutions. Ann. Appl. Probab. 21 (2011), no. 6, 2379--2423. doi:10.1214/11-AAP762.

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