A note on the integrability of the classical portfolio selection model

We revisit the classical Merton portfolio selection model from the perspective of integrability analysis. By an application of a nonlocal transformation the nonlinear partial differential equation for the two-asset model is mapped into a linear option valuation equation with a consumption dependent source term. This result is identical to the one obtained by Cox-Huang [J.C. Cox, C-f. Huang, Optimal consumption and portfolio policies when asset prices follow a diffusion process, J. Econom. Theory 49 (1989) 33-88], using measure theory and stochastic integrals. The nonlinear two-asset equation is then analyzed using the theory of Lie symmetry groups. We show that the linearization is directly related to the structure of the generalized symmetries. (C) 2010 Elsevier Ltd. All rights reserved.