Optimal Consumption and Portfolio with Ambiguity to Markovian Switching

This paper considers the problem of maximizing expectedutility from consumption and terminal wealth under model uncertainty for a generalsemimartingale market, where the agent with an initial capital and a random endowmentcan 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 investmentproblem exists. Then we deduce that the value functions of primal problem and dualproblem are convex conjugate functions. Furthermore we consider a diffusion-jump-modelwhere the coefficients depend on the state of a Markov chain and the investor isambiguity to the intensity of the underlying Poisson process. Finally, for an agentwith the logarithmic utility function, we use the stochastic control method to derivethe Hamilton-Jacobi-Bellmann (HJB) equation. And the solution to this HJB equation canbe determined numerically. We also show how thereby the optimal investment strategycan be computed.