Optimal Consumption And Portfolio Problem With Regimes Switching Under Knightian Uncertainty

The classical consumption and investment model is analyzed by Merton in the late1960s. Since then, this kind of problem has been one of the major research areas in mathematical finance. This paper deals with the problem of optimal consump-tion and investment under Knightian uncertainty. Here, we distinguish Knightian uncertainty and risk. Risk denotes the fact that the future development of the stock market is random but the agent knows which future scenarios are possible and knows their market measure. Knightian uncertainty denotes the fact that market participants usually do not know the market measure or at least not exactly. In this paper, the agent optimizes his consumption and investment strategy with respect to a robust utility functional.First, we consider the investment problem in a general semimartingale frame-work where the agent may invest in the stock market and receives additional random endowment.We find a solution to the investment problem by using the martingale method and the dual theory. We prove that under appropriate assumptions a unique solution to the investment problem exists and is characterized. Furthermore, we can deduce that the value functions of primal and dual problem are convex conjugate functions. After that, we consider a diffusion-jump-model where the coefficients de-pend on the state of a Markov chain and the investor is uncertain about the intensity of the underlying Poisson process. In this model we consider an agent with loga-rithmic utility function. We use the stochastic control method to derive the HJB equation. The solution to this HJB equation can be determined numerically and we show how thereby the optimal investment strategy can be computed.Second, we consider the consumption and investment problem in a general semimartingale market. Here, the agent can invest an initial capital and a random endowment. To find a solution to the consumption and investment problem we use the martingale method. We prove that under appropriate assumptions a unique solution to the investment problem exists and is characterized too. Furthermore, in a diffusion-jump-model we consider an agent with logarithmic and HARA utility function. For both we use the stochastic control method to derive the HJB equation. The solution to this HJB equation can be determined numerically and we show how thereby the optimal investment strategy can be computed.