Characterization of optimal dual measures via distortion

We derive representations for the optimal dual martingale measure Q∗ associated with the dual to a primal utility maximization problem in a class of incomplete diffusion models containing a traded stock and a non-traded correlated stochastic factor. Using a distortion power solution [29] for the primal problem, an explicit solution is obtained for the dual value function, yielding representations for the dual measure and a novel solution for a dual stochastic control problem. The optimal measure is recast as the q-optimal measure Q, with q ≤ 1 for HARA and CARA utilities, which we treat in a unified manner using the parameter q. We extend the Hobson [13] representation equation for Q to q < 1. A similar program is applied to a problem with random endowment under exponential preferences, yielding representations for indifference prices and the dual minimizer, and an extension of the Hobson representation equation to the problem with random endowment.