Robust stochastic control and equivalent martingale measures

Bernt Oksendal () and Agnès Sulem ()
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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

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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.

Keywords: robust control; model uncertainty; worst case scenario; portfolio optimization; Lévy Market (search for similar items in EconPapers)
Date: 2011-03-03
Note: View the original document on HAL open archive server: https://inria.hal.science/inria-00573117v1
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Published in [Research Report] RR-7557, INRIA. 2011, pp.11

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