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

TL;DR

This paper develops a duality framework for utility maximization problems with ratchet and drawdown constraints on consumption in incomplete semimartingale markets, extending the theory to include initial wealth and consumption bounds.

Contribution

It introduces a natural extension of the running maximum for optional processes and characterizes dual domains using solidity, advancing the understanding of constrained utility maximization.

Findings

Duality results for constrained utility maximization in incomplete markets.

Characterization of dual domains via solidity and ordering.

Detailed solutions in the complete market case.

Abstract

In this paper, we study 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 detailed characterization of solutions in the complete market case.