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SOLVABILITY AND NUMERICAL SIMULATION OF BSDEs RELATED TO BSPDEs WITH APPLICATIONS TO UTILITY MAXIMIZATION

Author

Listed:
  • PETER IMKELLER

    (Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany)

  • ANTHONY RÉVEILLAC

    (Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany)

  • JIANING ZHANG

    (Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany)

Abstract

In this paper we study BSDEs arising from a special class of backward stochastic partial differential equations (BSPDEs) that is intimately related to utility maximization problems with respect to arbitrary utility functions. After providing existence and uniqueness we discuss the numerical realizability. Then we study utility maximization problems on incomplete financial markets whose dynamics are governed by continuous semimartingales. Adapting standard methods that solve the utility maximization problem using BSDEs, we give solutions for the portfolio optimization problem which involve the delivery of a liability at maturity. We illustrate our study by numerical simulations for selected examples. As a byproduct we prove existence of a solution to a very particular quadratic growth BSDE with unbounded terminal condition. This complements results on this topic obtained in Briand and Hu (2006, 2008) and Briand et al. (2007).

Suggested Citation

  • Peter Imkeller & Anthony Réveillac & Jianing Zhang, 2011. "SOLVABILITY AND NUMERICAL SIMULATION OF BSDEs RELATED TO BSPDEs WITH APPLICATIONS TO UTILITY MAXIMIZATION," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(05), pages 635-667.
    • Handle: RePEc:wsi:ijtafx:v:14:y:2011:i:05:n:s0219024911006437
    DOI: 10.1142/S0219024911006437
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    References listed on IDEAS

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    1. M. Mania & R. Tevzadze, 2008. "Backward Stochastic PDEs related to the utility maximization problem," Papers 0806.0240, arXiv.org.
    2. Sara Biagini & Marco Frittelli & Matheus R. Grasselli, 2009. "Indifference price with general semimartingales," Papers 0905.4657, arXiv.org.
    3. Michael Mania & Revaz Tevzadze, 2008. "Backward Stochastic PDEs Related to the Utility Maximization Problem," ICER Working Papers - Applied Mathematics Series 07-2008, ICER - International Centre for Economic Research.
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    Cited by:

    1. Jun Moon, 2022. "State and Control Path-Dependent Stochastic Zero-Sum Differential Games: Viscosity Solutions of Path-Dependent Hamilton–Jacobi–Isaacs Equations," Mathematics, MDPI, vol. 10(10), pages 1-32, May.
    2. Horst, Ulrich & Hu, Ying & Imkeller, Peter & Réveillac, Anthony & Zhang, Jianing, 2014. "Forward–backward systems for expected utility maximization," Stochastic Processes and their Applications, Elsevier, vol. 124(5), pages 1813-1848.

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