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Portfolio optimization under model uncertainty and BSDE games

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  • Bernt Øksendal
  • Agnès Sulem

Abstract

We consider robust optimal portfolio problems for markets modeled by (possibly non-Markovian) Itô–Lévy processes. Mathematically, the situation can be described as a stochastic differential game, where one of the players (the agent) is trying to find the portfolio that 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 into a stochastic differential game for backward stochastic differential equations (a 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 Øksendal & Agnès Sulem, 2011. "Portfolio optimization under model uncertainty and BSDE games," Quantitative Finance, Taylor & Francis Journals, vol. 11(11), pages 1665-1674.
    • Handle: RePEc:taf:quantf:v:11:y:2011:i:11:p:1665-1674
    DOI: 10.1080/14697688.2011.615219
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    Citations

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    Cited by:

    1. Bernt {O}ksendal & Agn`es Sulem, 2013. "Dynamic robust duality in utility maximization," Papers 1304.5040, arXiv.org, revised Sep 2015.
    2. 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.
    3. Roger J. A. Laeven & Mitja Stadje, 2014. "Robust Portfolio Choice and Indifference Valuation," Mathematics of Operations Research, INFORMS, vol. 39(4), pages 1109-1141, November.
    4. 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.
    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. Yang Shen & Tak Kuen Siu, 2018. "A Risk-Based Approach for Asset Allocation with A Defaultable Share," Risks, MDPI, vol. 6(1), pages 1-27, February.
    7. 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.
    8. Junichi Imai, 2022. "A Numerical Method for Hedging Bermudan Options under Model Uncertainty," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 893-916, June.
    9. Calisto Guambe & Rodwell Kufakunesu, 2017. "Optimal investment-consumption and life insurance selection problem under inflation. A BSDE approach," Papers 1711.01760, arXiv.org.

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