Maximin investment problems for discounted and total wealth

We study an optimal investment problem for a diffusion market model such that the risk-free rate, the appreciation rates and the volatility of the stocks are all random; they are not necessarily adapted to the driving Brownian motion, and their distributions are unknown, but they are supposed to be currently observable. The optimal investment problem is stated as a problem with a maximin performance criterion which leads to maximization of the minimum of expected utility over all distributions of parameters. We found that including the non-discounted wealth to the utility (in addition to the discounted wealth) implies an additional condition for the saddle point of the maximin problem: it must be a solution of an additional minimization problem for the risk-free return. This is different from the case when the utility depends on the discounted wealth only. Using these results, the maximin problem is reduced to a linear parabolic equation and minimization (over two scalar parameters). It is an important addition to the result obtained in the author's paper (2006).