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An elementary approach to the Merton problem
Author(s) -
Herdegen Martin,
Hobson David,
Jerome Joseph
Publication year - 2021
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/mafi.12311
Subject(s) - mathematical economics , mathematical proof , constant (computer programming) , bellman equation , simple (philosophy) , mathematics , argument (complex analysis) , mathematical optimization , computer science , philosophy , biochemistry , chemistry , geometry , epistemology , programming language
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.