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Robust stochastic control and equivalent martingale measures

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 nationale des ponts et chaussées - UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12)

Abstract

We study a class of robust, or worst case scenario, optimal control problems for jump diffusions. The scenario is represented by a probability measure equivalent to the initial probability law. We show that if there exists a control that annihilates the noise coefficients in the state equation and a scenario which is an equivalent martingale measure for a specific process which is related to the control-derivative of the state process, then this control and this probability measure are optimal. We apply the result to the problem of consumption and portfolio optimization under model uncertainty in a financial market, where the price process S(t) of the risky asset is modeled as a geometric Itô-Lévy process. In this case the optimal scenario is an equivalent local martingale measure of S(t). We solve this problem explicitly in the case of logarithmic utility functions.

Suggested Citation

  • Bernt Oksendal & Agnès Sulem, 2011. "Robust stochastic control and equivalent martingale measures," Working Papers inria-00573117, HAL.
    • Handle: RePEc:hal:wpaper:inria-00573117
    Note: View the original document on HAL open archive server: https://inria.hal.science/inria-00573117v1
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    References listed on IDEAS

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    1. Lars Peter Hansen & Thomas J Sargent, 2014. "Robust Control and Model Uncertainty," World Scientific Book Chapters, in: UNCERTAINTY WITHIN ECONOMIC MODELS, chapter 5, pages 145-154, World Scientific Publishing Co. Pte. Ltd..
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