Symmetry Reductions of a Hamilton-Jacobi-Bellman Equation Arising in Financial Mathematics

Abstract

We determine the solutions of a nonlinear Hamilton-Jacobi-Bellman equation which arises in the modelling of mean-variance hedging subject to a terminal condition.Firstly we establish those forms of the equation which admit the maximal number of Lie point symmetries and then examine each in turn.We show that the Lie method is only suitable for an equation of maximal symmetry.We indicate the applicability of the method to cases in which the parametric function depends also upon the time.