A note on the integrability of the classical portfolio selection model
type: Item
instance of: scholarly article
title: A note on the integrability of the classical portfolio selection model
zbMATH Open document ID: 1195.91147
author: J. Martínez
published in: Applied Mathematics Letters
publication date: 2010-08-18T00:00:00Z
review text: The authors study the classical Merton portfolio selection model and, more concretely, the problem of the optimal behaviour of an investor who wishes to maximize lifetime utility of consumption, plus a final bequest, subject to wealth, invested in a portfolio of assets modelled as geometric Brownian motions. \textit{R. C. Merton} [J. Econ. Theory 3, No. 4, 373--423 (1971; Zbl 1011.91502)] derived a nonlinear differential equation from the optimal control problem, while \textit{J. C. Cox} and \textit{C.-f. Huang} [J. Econ. Theory 49, No. 1, 33--83 (1989; Zbl 0678.90011)] obtained a linear differential equation. In this article, the authors map the nonlinear partial differential equation for the two-asset model into a linear option valuation equation with a consumption dependent source term, showing equivalence of both solutions by means of a generalized transformation on the optimal controls. This transformation is studied in the context of Lie symmetry groups for differential equations, showing how the generalized infinitesimal symmetries of Merton's equation contain the solution given by Cox and Huang inside the symmetry structure. Precisely, using this symmetry structure, the authors can construct the linearization of Merton's equation.
Mathematics Subject Classification ID: 91G10
Mathematics Subject Classification ID: 91G80
Mathematics Subject Classification ID: 17B80
Mathematics Subject Classification ID: 70G65
zbMATH DE Number: 5772547
zbMATH Keywords: Lie symmetry analysis
zbMATH Keywords: portfolio selection theory
zbMATH Keywords: Merton portfolio selection model
zbMATH Keywords: integrability analysis
zbMATH Keywords: partial differential equations
reviewed by: Angel Francisco Tenorio Villalon
MaRDI profile type: Publication
full work available at URL: j.aml.2010.04.046
OpenAlex ID: W2008108307
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DOI: 10.1016/J.AML.2010.04.046
description: scientific article; zbMATH DE number 5772547