Robust Optimal Control for a Consumption-investment Problem

We give an explicit PDE characterization for the solution of the problem of maximizing the utility of both terminal wealth and intertemporal consumption under model uncertainty. The underlying market model consists of a risky asset, whose volatility and long-term trend are driven by an external stochastic factor process. The robust utility functional is defined in terms of a HARA utility function with risk aversion parameter 0 < a < 1 and a dynamically consistent coherent risk measure, which allows for model uncertainty in the distributions of both the asset price dynamics and the factor process. Our method combines recent results by Wittmüss (2007) on the duality theory of robust optimization of consumption with a stochastic control approach to the dual problem of determining a 'worst-case martingale measure'.