On Optimal Retirement (How to Retire Early)

TL;DR

This paper formulates and solves an optimal control problem for early retirement investment strategies, identifying the optimal investment function to minimize retirement time under stochastic market dynamics.

Contribution

It introduces a new model for retirement investing using stochastic control and derives the optimal investment strategy among all nonanticipative strategies.

Findings

Derived the optimal investment function $f_0$ for early retirement.

Proved the optimality of $f_0$ among all nonanticipative strategies.

Provided a mathematical framework for early retirement planning under uncertainty.

Abstract

We pose an optimal control problem arising in a perhaps new model for retirement investing. Given a control function $f$ and our current net worth as $X(t)$ for any $t$, we invest an amount $f(X(t))$ in the market. We need a fortune of $M$ "superdollars" to retire and want to retire as early as possible. We model our change in net worth over each infinitesimal time interval by the Ito process $dX(t)= (1+f(X(t))dt+ f(X(t))dW(t)$. We show how to choose the optimal $f=f_0$ and show that the choice of $f_0$ is optimal among all nonanticipative investment strategies, not just among Markovian ones.