Dynamic Robust Duality in Utility Maximization.

Summary A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities: (i) The optimal terminal wealth X^*(T) : = X_{\varphi ^*}(T) of the problem to maximize the expected U-utility of the terminal wealth X_{\varphi }(T) generated by admissible portfolios \varphi (t). 0 \le t \le T in a market with the risky asset price process modeled as a semimartingale. (ii) The optimal scenario \frac{dQ^*}{dP} of the dual problem to minimize the expected V-value of \frac{dQ}{dP} over a family of equivalent local martingale measures Q, where V is the convex conjugate function of the concave function U. In this paper we consider markets modeled by Itô-Lévy processes. In the first part we use the maximum principle in stochastic control theory to extend the above relation to a dynamic relation, valid for all t \in [0,T]. We prove in particular that the optimal adjoint process for the primal problem coincides with the optimal density process, and that the optimal adjoint process for the dual problem coincides with the optimal wealth process.