Considering a representative agent in the market, we study the long-term optimal investment problem in a discrete-time financial market, introducing a set of restrictions in the admissible strategies. The drawdown constraints limit the size of possible losses of the portfolio and impose a floor-based performance measure. The optimal growth rate is characterized, and under suitable hypotheses it is proved that an optimal strategy exists. The approach to solving this problem is based on dynamic programming techniques and a fixed point argument adapted from the theory of Markov decision processes.

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.

The maximization of the long-term growth rate of expected utility is considered under drawdown constraints. In a general situation, the value and the optimal strategy of the problem are related to those of another ‘standard’ risk-sensitive-type portfolio optimization problem. Furthermore, an upside-chance maximization problem of a large deviation probability is stated as a ‘dual’ optimization problem. As an example, a ‘linear-quadratic’ model is studied in detail: the conditions to ensure the solvabilities of the problems are discussed and explicit expressions for the solutions are presented.