On Optimal Terminal Wealth Problems with Random Trading Times and Drawdown Constraints

We consider an investment problem where observing and trading are only possible at random times. In addition, we introduce drawdown constraints which require that the investor's wealth does not fall under a prior fixed percentage of its running maximum. The financial market consists of a riskless bond and a stock which is driven by a Lévy process. Moreover, a general utility function is assumed. In this setting we solve the investment problem using a related limsup Markov decision process. We show that the value function can be characterized as the unique fixed point of the Bellman equation and verify the existence of an optimal stationary policy. Under some mild assumptions the value function can be approximated by the value function of a contracting Markov decision process. We are able to use Howard's policy improvement algorithm for computing the value function as well as an optimal policy. These results are illustrated in a numerical example.

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