Explicit Optimal Solution in Maximin Setting for Investment Problems with Totally Unhedgeable Coefficients

We study optimal investment problem for a market model where the evolution of risky assets is described by Ito's equations. The risk-free rate, the appreciation rates, and the volatility of the stocks are all random; they are not necessary adapted to the driving Brownian motion, their distributions are unknown, and they are supposed to be currently observable. The optimal investment problem is stated as a problem with a maximin performance criterion to ensure that a strategy is found such that the minimum of expected utility over all possible parameters is maximal. We show that a saddle point exists and can be found via minimization over a single scalar parameter. The {\it maximin} problem is solved for a very general case via solution of a linear parabolic equation with explicit fundamental solution