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 Ito-L00E9vy 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.
Mathematics Subject Classification (2010). 93E20, 60G51, 60H20, 60G44, 91G80.
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References
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Øksendal, B., Sulem, A. (2011). Robust Stochastic Control and Equivalent Martingale Measures. In: Kohatsu-Higa, A., Privault, N., Sheu, SJ. (eds) Stochastic Analysis with Financial Applications. Progress in Probability, vol 65. Springer, Basel. https://doi.org/10.1007/978-3-0348-0097-6_12
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DOI: https://doi.org/10.1007/978-3-0348-0097-6_12
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