A Certainty Equivalent Merton Problem

TL;DR

This paper introduces a deterministic certainty equivalent formulation of the Merton problem, transforming it into a convex optimization problem that simplifies solving extensions and supports model predictive control.

Contribution

It identifies a certainty equivalent problem for the Merton problem and its extensions, enabling easier solution via convex optimization techniques.

Findings

The certainty equivalent problem shares the same optimal value and policy as the original.

Discretizing time converts the problem into a second-order cone program (SOCP).

The approach facilitates model predictive control for complex extensions.

Abstract

The Merton problem is the well-known stochastic control problem of choosing consumption over time, as well as an investment mix, to maximize expected constant relative risk aversion (CRRA) utility of consumption. Merton formulated the problem and provided an analytical solution in 1970; since then a number of extensions of the original formulation have been solved. In this note we identify a certainty equivalent problem, i.e., a deterministic optimal control problem with the same optimal value function and optimal policy, for the base Merton problem, as well as a number of extensions. When time is discretized, the certainty equivalent problem becomes a second-order cone program (SOCP), readily formulated and solved using domain specific languages for convex optimization. This makes it a good starting point for model predictive control, a policy that can handle extensions that are either too cumbersome or impossible to handle exactly using standard dynamic programming methods.