A problem of optimal switching and singular control with discretionary stopping in portfolio selection

TL;DR

This paper models an economic agent's portfolio and retirement decision as a dual zero-sum game involving optimal switching, singular control, and stopping, providing a solution to the associated Hamilton-Jacobi-Bellman quasi-variational inequality.

Contribution

It transforms a complex portfolio optimization problem into a dual game framework and derives the solution and optimal strategies using HJBQV analysis.

Findings

Derived the HJBQV for the dual game

Provided a verification theorem for the solution

Established duality between primal and dual problems

Abstract

In this paper we study the optimization problem of an economic agent who chooses a job and the time of retirement as well as consumption and portfolio of assets. The agent is constrained in the ability to borrow against future income. We transform the problem into a dual two-person zero-sum game, which involves a controller, who is a minimizer and chooses a non-increasing process, and a stopper, who is a maximizer and chooses a stopping time. We derive the Hamilton-Jacobi- Bellman quasi-variational inequality(HJBQV) of a max-min type arising from the game. We provide a solution to the HJBQV and verification that it is the value of the game. We establish a duality result which allows to derive the optimal strategies and value function of the primal problem from those of the dual problem.