Optimal Investment Stopping Problem with Nonsmooth Utility in Finite Horizon

TL;DR

This paper develops a novel approach to solve an optimal stopping problem with nonsmooth utility in finite horizon, using dual transformations to convert a nonlinear free boundary problem into a linear one, enabling analysis of optimal strategies.

Contribution

It introduces a new methodology employing dual transformations to analyze a complex mixed control and stopping problem with nonsmooth utility, which differs from existing methods.

Findings

Derived properties of the optimal stopping boundary.

Established the structure of the optimal investment strategy.

Provided a systematic way to solve nonlinear free boundary problems.

Abstract

In this paper, we investigate an interesting and important stopping problem mixed with stochastic controls and a \textit{nonsmooth} utility over a finite time horizon. The paper aims to develop new methodologies, which are significantly different from those of mixed dynamic optimal control and stopping problems in the existing literature, to figure out a manager's decision. We formulate our model to a free boundary problem of a fully \textit{nonlinear} equation. By means of a dual transformation, however, we can convert the above problem to a new free boundary problem of a \textit{linear} equation. Finally, using the corresponding inverse dual transformation, we apply the theoretical results established for the new free boundary problem to obtain the properties of the optimal strategy and the optimal stopping time to achieve a certain level for the original problem over a finite time investment horizon.