Title:Utility maximization with ratchet and drawdown constraints on consumption in incomplete semimartingale markets

Abstract:This paper studies expected utility maximization under ratchet and drawdown constraints on consumption in a general incomplete semimartingale market using duality methods. The optimization is considered with respect to two parameters: the initial wealth and the essential lower bound on consumption process. In order to state the problem and define the primal domains, we introduce a natural extension of the notion of running maximum to arbitrary non-negative optional processes and study its properties. The dual domains for optimization are characterized in terms of solidity with respect to an ordering that is introduced on the set of non-negative optional processes. The abstract duality result we obtain for the optimization problem is used in order to derive a more specific characterization of solution in case of ratchet constraint in a complete market. The method described in the paper might be useful for other control problems involving running maxima or increasing controls.