Robust Stochastic Control and Equivalent Martingale Measures

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.