An elementary approach to the Merton problem. (6th May 2021)

• Record Type: Journal Article
• Authors: Herdegen, Martin; Hobson, David; Jerome, Joseph
• Other Names: Obloj Jan guestEditor.; Zariphopoulou Thaleia guestEditor.
• Is Part Of: Mathematical finance. Volume 31:Number 4(2021)
• Page Start: 1218
• Page End: 1239
• Publication Date: 2021-05-06
• Subjects: investment/consumption -- Merton problem -- primal approach -- verification argument; Business mathematics -- Periodicals; 332
• DOI: 10.1111/mafi.12311
• Languages: English
• ISSNs: 0960-1627

Abstract

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.