Maximin investment problems for discounted and total wealth

Abstract

We study an optimal investment problem for a continuous-time incomplete 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 in 'maximin' setting which leads to maximization of the minimum of expected utility over all distributions of parameters. We found that the presence of the non-discounted wealth in the performance criterion (in addition to the discounted wealth) implies an additional condition for the saddle point of the maximin problem: the saddle point must include the minimum of the possible risk-free return. This is different from the case when the utility depends on the discounted wealth only. Using this result, the maximin problem is reduced to a linear parabolic equation and minimization over two scalar parameters. It is an important development of the results obtained in Dokuchaev (2002, Dynamic Portfolio Strategies: Quantitative Methods and Empirical Rules for Incomplete Information. Boston: Kluwer; 2006, IMA J. Manage. Math., 17, 257-276).