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Solutions of BSDE’s with jumps and quadratic/locally Lipschitz generator

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  • Antonelli, Fabio
  • Mancini, Carlo

Abstract

Inspired by utility optimization problems in finance, in this paper we prove the existence of the solution of a class of BSDE’s driven by a Brownian motion and a jump process, whose generator shows quadratic growth in the Brownian component and exponential growth with respect to the jump term. Existence and uniqueness of the solution is established first for bounded terminal value, then we extend the existence result to the unbounded case, under appropriate hypotheses.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:10:p:3124-3144
    DOI: 10.1016/j.spa.2016.04.004
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    References listed on IDEAS

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    1. Marie-Amélie Morlais, 2009. "Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem," Finance and Stochastics, Springer, vol. 13(1), pages 121-150, January.
    2. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
    3. Claudia Ceci & Anna Gerardi, 2011. "Utility indifference valuation for jump risky assets," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 34(2), pages 85-120, November.
    4. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    5. 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.
    6. 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.
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    Citations

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    Cited by:

    1. Fujii, Masaaki & Takahashi, Akihiko, 2018. "Quadratic–exponential growth BSDEs with jumps and their Malliavin’s differentiability," Stochastic Processes and their Applications, Elsevier, vol. 128(6), pages 2083-2130.
    2. Masaaki Fujii & Akihiko Takahashi, 2017. "Anticipated Backward SDEs with Jumps and quadratic-exponential growth drivers," Papers 1705.02440, arXiv.org, revised Jul 2018.
    3. Masaaki Fujii & Akihiko Takahashi, 2017. "Anticipated Backward SDEs with Jumps and quadratic-exponential growth drivers," CARF F-Series CARF-F-409, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    4. Fabio Antonelli & Carlo Mancini, 2016. "Consumption optimization for recursive utility in a jump-diffusion model," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 39(2), pages 293-310, November.
    5. Masaaki Fujii & Akihiko Takahashi, 2016. "Quadratic-exponential growth BSDEs with Jumps and their Malliavin’s Differentiability (revised version of CARF-F-376)," CARF F-Series CARF-F-395, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    6. Masaaki Fujii & Akihiko Takahashi, 2017. "Quadratic-exponential growth BSDEs with Jumps and their Malliavin's Differentiability (Forthcoming in Stochastic Processes and their Applications)(revised version of CARF-F-395)," CARF F-Series CARF-F-420, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    7. Masaaki Fujii & Akihiko Takahashi, 2017. "Anticipated Backward SDEs with Jumps and Quadratic-Exponential Growth Drivers," CIRJE F-Series CIRJE-F-1047, CIRJE, Faculty of Economics, University of Tokyo.
    8. Marina Santacroce & Paola Siri & Barbara Trivellato, 2023. "Forward Backward SDEs Systems for Utility Maximization in Jump Diffusion Models," Papers 2302.08253, arXiv.org.
    9. Masaaki Fujii & Akihiko Takahashi, 2018. "Anticipated Backward SDEs with Jumps and quadratic-exponential growth drivers (Revised version of F-409)," CARF F-Series CARF-F-431, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    10. Masaaki Fujii & Akihiko Takahashi, 2017. "Anticipated Backward SDEs with Jumps and Quadratic-Exponential Growth Drivers," CIRJE F-Series CIRJE-F-1047, CIRJE, Faculty of Economics, University of Tokyo.

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    Keywords

    BSDE’s; Jump–diffusion processes;

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