Optimal Consumption and Portfolio with Ambiguity to Markovian Switching

This paper considers the problem of maximizing expected utility from consumption and terminal wealth under model uncertainty for a general semimartingale market, where the agent with an initial capital and a random endowment can invest. To find a solution to the investment problem we use the martingale method. We first prove that under appropriate assumptions a unique solution to the investment problem exists. Then we deduce that the value functions of primal problem and dual problem are convex conjugate functions. Furthermore we consider a diffusion-jump-model where the coefficients depend on the state of a Markov chain and the investor is ambiguity to the intensity of the underlying Poisson process. Finally, for an agent with the logarithmic utility function, we use the stochastic control method to derive the Hamilton-Jacobi-Bellmann (HJB) equation. And the solution to this HJB equation can be determined numerically. We also show how thereby the optimal investment strategy can be computed.