An Elementary Approach to the Merton Problem

Paper accepted to Mathematical Finance.

21 Pages Posted: 1 Jul 2020 Last revised: 30 Mar 2021

See all articles by Martin Herdegen

Martin Herdegen

University of Warwick - Department of Statistics

David Hobson

University of Warwick

Joseph Jerome

University of Warwick - Department of Statistics

Date Written: March 30, 2021

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.

Keywords: Mathematical Finance, Merton Problem, Stochastic Control, Expected Utility Maximization, Numeraire Change

JEL Classification: G11, C61, A23

Suggested Citation

Herdegen, Martin and Hobson, David and Jerome, Joseph, An Elementary Approach to the Merton Problem (March 30, 2021). Paper accepted to Mathematical Finance., Available at SSRN: https://ssrn.com/abstract=3623047 or http://dx.doi.org/10.2139/ssrn.3623047

Martin Herdegen

University of Warwick - Department of Statistics ( email )

Coventry CV4 7AL
United Kingdom

David Hobson

University of Warwick ( email )

CV4 7AL
United Kingdom

Joseph Jerome (Contact Author)

University of Warwick - Department of Statistics ( email )

Coventry, CV47AL
United Kingdom

HOME PAGE: http://https://warwick.ac.uk/fac/sci/statistics/staff/research_students/jerome/

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