An elementary approach to the Merton problem

TL;DR

This paper presents a simple and elegant method to solve the infinite-horizon Merton investment-consumption problem in a Black-Scholes-Merton market, overcoming previous technical restrictions with a stochastic utility perturbation approach.

Contribution

It introduces a novel, elementary approach using stochastic utility perturbation to address complexities in the Merton problem beyond standard stochastic control methods.

Findings

The method simplifies the verification process for the Merton problem.

It removes the need for parameter restrictions like R<1.

The approach broadens the applicability of solution techniques in continuous-time finance.

Abstract

In this article we consider the infinite-horizon Merton investment-consumption problem in a constant-parameter Black - Scholes - Merton market for an agent with constant relative risk aversion R. The classical primal approach is to write down a candidate value function and to use a verification argument to prove that this is the solution to the problem. However, features of the problem take it outside the standard settings of stochastic control, and the existing primal verification proofs rely on parameter restrictions (especially, but not only, R<1), restrictions on the space of admissible strategies, or intricate approximation arguments. The purpose of this paper is to show that these complications can be overcome using a simple and elegant argument involving a stochastic perturbation of the utility function.