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Anticipated Backward SDEs with Jumps and quadratic-exponential growth drivers

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Abstract

In this paper, we study a class of Anticipated Backward Stochastic Differential Equations (ABSDE) with jumps. The solution of the ABSDE is a triple (Y,Z, ψ) where Y is a semimartingale, and (Z, ψ) are the diffusion and jump coefficients. We allow the driver of the ABSDE to have linear growth on the uniform norm of Y ’s future paths, as well as quadratic and exponential growth on the spot values of (Z, ψ), respectively. The existence of the unique solution is proved for Markovian and non-Markovian settings with different structural assumptions on the driver. In the former case, some regularities on (Z, ψ) with respect to the forward process are also obtained.
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