Optimal Investment with Stopping in Finite Horizon

TL;DR

This paper develops new methods for solving finite-horizon stochastic control and optimal stopping problems, transforming complex nonlinear free boundary problems into linear ones, with applications in wealth management.

Contribution

It introduces a novel dual transformation approach to analyze finite-horizon control and stopping problems, providing practical solutions for wealth management strategies.

Findings

Derived properties of optimal investment strategies.

Identified optimal stopping times for wealth targets.

Transformed nonlinear free boundary problems into linear equations.

Abstract

In this paper, we investigate dynamic optimization problems featuring both stochastic control and optimal stopping in 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 study a manager's decision. We formulate our model to a free boundary problem of a fully nonlinear equation. Furthermore, by means of a dual transformation for the above problem, we convert the above problem to a new free boundary problem of a linear equation. Finally, we apply the theoretical results to challenging, yet practically relevant and important, risk-sensitive problems in wealth management to obtain the properties of the optimal strategy and the right time to achieve a certain level over a finite time investment horizon.