Retirement decision with addictive habit persistence in a jump diffusion market

TL;DR

This paper models optimal retirement, investment, and consumption strategies in a jump diffusion market with habit persistence, revealing how wealth, habits, and wages influence retirement timing and strategy discontinuities.

Contribution

It introduces a duality approach with habit reduction to characterize the retirement boundary in a jump diffusion setting, including technical advancements in integral equation proofs.

Findings

Immediate retirement occurs when wealth exceeds a critical wage proportion.

Jump risks can induce discontinuous investment strategies.

Numerical results illustrate parameter effects on retirement decisions.

Abstract

This paper investigates the optimal retirement decision, investment, and consumption strategies in a market with jump diffusion, taking into account habit persistence and stock-wage correlation. Our analysis considers multiple stocks and a finite time framework, intending to determine the retirement boundary of the ``wealth-habit-wage" triplet $(x, h, w)$. To achieve this, we use the habit reduction method and a duality approach to obtain the retirement boundary of the primal variables and feedback forms of optimal strategies. { When dealing with the dual problem, we address technical challenges in the proof of integral equation characterization of optimal retirement boundary using a $C^1$ version of It$\hat{\rm o}$'s formula.} Our results show that when the so-called ``de facto wealth" exceeds a critical proportion of wage, an immediate retirement is the optimal choice for the agent. Additionally, we find that the introduction of jump risks allows for the possibility of discontinuous investment strategies within the working region, which is a novel and insightful finding. Our numerical results effectively illustrate these findings by varying the parameters.