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

We study optimal investment problem for a market model where the evolution of risky assets prices is described by Itô's 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 ‘maximin’ 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 a solution of the standard 1D 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.