A note on the integrability of the classical portfolio selection model
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.
scientific article; zbMATH DE number 5772547