Abstract
AbstractIn 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, ), 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.
Highlights
AND OVERVIEWIn the Merton investment-consumption problem (Merton, 1969, 1971) an agent seeks to maximize the expected integrated discounted utility of consumption over the infinite horizon in a model with a risky asset and a riskless bond
When parameters are constant and the utility function is of power type, it is straightforward to write down the candidate value function
It is difficult to give a concise, rigorous verification proof via analysis of the value function, and many textbooks either finesse the issues or restrict attention to a subclass of admissible strategies, and/or restrict attention to a subset of parameter combinations. The need for such a verification argument has been obviated by the development of proofs using the dual method, which provides a powerful and intuitive alternative approach, see Biagini (2010) for a survey
Summary
In the Merton investment-consumption problem (Merton, 1969, 1971) an agent seeks to maximize the expected integrated discounted utility of consumption over the infinite horizon in a model with a risky asset and a riskless bond. In a series of appendices we: first, give an example which illustrates how one of the clearly-stated assumptions may fail; second, give a small amount of detail on the Karatzas et al (1986) and Davis and Norman (1990) approaches to the verification problem; third, discuss the case of logarithmic utility; fourth, consider the Merton problem under a change of numéraire and discuss the role of the parameter δ; and fifth, for completeness, give a brief discussion of duality methods for the Merton problem. Our proof is an improvement on the existing primal results in at least three important ways It places no restrictions on the class of admissible strategies: for example, unlike much of the stochastic control literature, it does not require the fraction of wealth invested in the risky asset to be bounded. Our proof is simple, elegant and concise and not counting the derivation of the candidate solution and candidate value function can be written up in just over one page (Theorem 5.1 and Corollary 5.4)
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