A Robustness Result for Stochastic Control

The solution of a stochastic control problem depends on the underlying model. The actual real world model may not be known precisely and so one solves the problem for a hypothetical model, that is in general different but close to the real one; the optimal (or nearly optimal) control of the hypothetical model is then used as solution for the real problem. In this paper we assume that, what is not precisely known, is the underlying probability measure that determines the distribution of the random quantities driving the model. We investigate two ways to derive a bound on the suboptimality of the optimal control of the hypothetical problem when this control is used in the real problem. Both bounds are in terms of the Radon-Nikodym derivative of the underlying real world measure with respect to the hypothetical one. We finally investigate how the bounds compare to each other.