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Portfolio optimization under model uncertainty and BSDE games

Author

Listed:
  • Bernt Oksendal

    (CMA - Center of Mathematics for Applications [Oslo] - Department of Mathematics [Oslo] - Faculty of Mathematics and Natural Sciences [Oslo] - UiO - University of Oslo)

  • Agnès Sulem

    (MATHFI - Financial mathematics - Inria Paris-Rocquencourt - Inria - Institut National de Recherche en Informatique et en Automatique - ENPC - École des Ponts ParisTech - UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12)

Abstract

We consider some robust optimal portfolio problems for markets modeled by (possibly non-Markovian) jump diffusions. Mathematically the situation can be described as a stochastic differential game, where one of the players (the agent) is trying to find the portfolio which maximizes the utility of her terminal wealth, while the other player ("the market") is controlling some of the unknown parameters of the market (e.g. the underlying probability measure, representing a model uncertainty problem) and is trying to minimize this maximal utility of the agent. This leads to a worst case scenario control problem for the agent. In the Markovian case such problems can be studied using the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation, but these methods do not work in the non-Markovian case. We approach the problem by transforming it to a stochastic differential game for backward differential equations (BSDE game). Using comparison theorems for BSDEs with jumps we arrive at criteria for the solution of such games, in the form of a kind of non-Markovian analogue of the HJBI equation. The results are illustrated by examples.

Suggested Citation

  • Bernt Oksendal & Agnès Sulem, 2011. "Portfolio optimization under model uncertainty and BSDE games," Working Papers inria-00570532, HAL.
    • Handle: RePEc:hal:wpaper:inria-00570532
    Note: View the original document on HAL open archive server: https://inria.hal.science/inria-00570532
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    References listed on IDEAS

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    1. Jan Ubøe & Bernt Øksendal & Knut Aase & Nicolas Privault, 2000. "White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance," Finance and Stochastics, Springer, vol. 4(4), pages 465-496.
    2. 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.
    3. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
    4. M. Mania & R. Tevzadze, 2008. "Backward Stochastic PDEs related to the utility maximization problem," Papers 0806.0240, arXiv.org.
    5. Michael Mania & Martin Schweizer, 2005. "Dynamic exponential utility indifference valuation," Papers math/0508489, arXiv.org.
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    1. Bernt {O}ksendal & Agn`es Sulem, 2013. "Dynamic robust duality in utility maximization," Papers 1304.5040, arXiv.org, revised Sep 2015.
    2. Bernt Øksendal & Agnès Sulem, 2014. "Forward–Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty," Journal of Optimization Theory and Applications, Springer, vol. 161(1), pages 22-55, April.
    3. Siu, Tak Kuen, 2016. "A functional Itô’s calculus approach to convex risk measures with jump diffusion," European Journal of Operational Research, Elsevier, vol. 250(3), pages 874-883.
    4. Calisto Guambe & Rodwell Kufakunesu, 2017. "Optimal investment-consumption and life insurance selection problem under inflation. A BSDE approach," Papers 1711.01760, arXiv.org.
    5. 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.
    6. Peng, Xingchun & Chen, Fenge & Hu, Yijun, 2014. "Optimal investment, consumption and proportional reinsurance under model uncertainty," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 222-234.

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