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Anticipated Backward SDEs with Jumps and quadratic-exponential growth drivers

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  • Masaaki Fujii
  • Akihiko Takahashi

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 $(Y,Z,\psi)$ where $Y$ is a semimartingale, and $(Z,\psi)$ 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 $(Z,\psi)$, 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 $(Z,\psi)$ with respect to the forward process are also obtained.

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  • Masaaki Fujii & Akihiko Takahashi, 2017. "Anticipated Backward SDEs with Jumps and quadratic-exponential growth drivers," Papers 1705.02440, arXiv.org, revised Jul 2018.
    • Handle: RePEc:arx:papers:1705.02440
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    References listed on IDEAS

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    1. Masaaki Fujii & Akihiko Takahashi, 2015. "Quadratic-exponential Growth BSDEs with Jumps and their Malliavin's Differentiability," CIRJE F-Series CIRJE-F-997, CIRJE, Faculty of Economics, University of Tokyo.
    2. Morlais, Marie-Amelie, 2010. "A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1966-1995, September.
    3. Antonelli, Fabio & Mancini, Carlo, 2016. "Solutions of BSDE’s with jumps and quadratic/locally Lipschitz generator," Stochastic Processes and their Applications, Elsevier, vol. 126(10), pages 3124-3144.
    4. Olivier Menoukeu Pamen, 2015. "Optimal Control for Stochastic Delay Systems Under Model Uncertainty: A Stochastic Differential Game Approach," Journal of Optimization Theory and Applications, Springer, vol. 167(3), pages 998-1031, December.
    5. Royer, Manuela, 2006. "Backward stochastic differential equations with jumps and related non-linear expectations," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1358-1376, October.
    6. Masaaki Fujii & Akihiko Takahashi, 2015. "Quadratic-exponential growth BSDEs with Jumps and their Malliavin’s Differentiability," CARF F-Series CARF-F-376, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
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