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Discrete time approximation of decoupled Forward-Backward SDE with jumps

By Bruno Bouchard and Romuald Elie

Abstract

International audienceWe study a discrete-time approximation for solutions of systems of decoupled forward-backward stochastic differential equations with jumps. Assuming that the coefficients are Lipschitz-continuous, we prove the convergence of the scheme when the number of time steps $n$ goes to infinity. When the jump coefficient of the first variation process of the forward component satisfies a non-degeneracy condition which ensures its inversibility, we obtain the optimal convergence rate $n^{-1/2}$. The proof is based on a generalization of a remarkable result on the path-regularity of the solution of the backward equation derived by Zhang (2001, 2004) in the no-jump case. A similar result is obtained without the non-degeneracy assumption whenever the coefficients are $C^1_b$ with Lipschitz derivatives. Several extensions of these results are discussed. In particular, we propose a convergent scheme for the resolution of systems of coupled semilinear parabolic PDE's

Topics: Malliavin calculus, Discrete-time approximation, forward-backward SDE's with jumps, Malliavin calculus., MSC Classification (2000): 65C99, 60H07, 60J75., [MATH.MATH-PR] Mathematics [math]/Probability [math.PR]
Publisher: Elsevier
Year: 2008
OAI identifier: oai:HAL:hal-00015486v1
Provided by: Hal-Diderot
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