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Discrete time approximation of decoupled Forward-Backward SDE with jumps

Abstract : We 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.
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Submitted on : Thursday, December 8, 2005 - 3:34:46 PM
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Bruno Bouchard, Romuald Elie. Discrete time approximation of decoupled Forward-Backward SDE with jumps. Stochastic Processes and their Applications, Elsevier, 2008, 118 (1), pp.53-75. ⟨hal-00015486⟩

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