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.

Optimal consumption under a drawdown constraint over a finite horizon

“…Jeon and Park (2021) further extend the work in Arun (2012) by considering general utility functions using the martingale approach and the dual formulation to some optimal stopping problems over an infinite horizon. Tanana (2021) employs the general duality approach and establishes the existence of the optimal consumption under a drawdown constraint in incomplete semimartingale markets. Jeon and Oh (2022) recently generalizes the approach in Jeon and Park (2021) to the model with a finite horizon and address the existence of solution to the dual optimal stopping problem.…”