Saddle points for maximin investment problems with observable but non-predictable parameters: solution via heat equation.

Abstract

We study optimal investment problem for a market model where the evolution of risky assets prices is described by Itoˆs equations. The risk-free rate, the appreciation rates, and the volatility of the stocks are all random; they depend on a random parameter that is not adapted to the driving Brownian motion. The distribution of this parameter is unknown. The optimal investment problem is stated in a mazimin setting to ensure that a strategy is found such that the minimum of expected utility over all possible distributions of parameters is maximal. We show that a saddle point exists and can be found via solution of the standard one-dimensional heat equation with a Cauchy condition defined via one-dimensional minimization. This solution even covers models with unknown solution for a given distribution of the market parameters.