Convex duality for stochastic singular control problems

TL;DR

This paper develops a convex duality framework for stochastic singular control problems, extending existing theories to include optimal expected utility from increasing controls, with applications in finance and investment.

Contribution

It formulates a duality theory for singular control problems, identifying dual functionals and establishing full duality results for primal and dual value functions.

Findings

Established a duality framework for singular control problems.

Identified the dual functional and proved full duality.

Applied results to investment and utility maximization problems.

Abstract

We develop a general theory of convex duality for certain singular control problems, taking the abstract results by Kramkov and Schachermayer (1999) for optimal expected utility from nonnegative random variables to the level of optimal expected utility from increasing, adapted controls. The main contributions are the formulation of a suitable duality framework, the identification of the problem's dual functional as well as the full duality for the primal and dual value functions and their optimizers. The scope of our results is illustrated by an irreversible investment problem and the Hindy-Huang-Kreps utility maximization problem for incomplete financial markets.