Maximin Criterion for Observable but Nonpredictable Parameters

In this chapter, it is assumed that the risk-free rate, the appreciation rates, and the volatility rates of the stocks are all random; they are not adapted to the driving Brownian motion, and their distributions are unknown, but they are currently observable. Admissible strategies are based on current observations of the stock prices and the aforementioned parameters. The optimal investment problem is stated as a problem with a maximin performance criterion. This criterion is to ensure that a strategy is found such that the minimum of utility over all distributions of parameters is maximal. It is shown that the duality theorem holds for the problem and that the maximin problem is reduced to the minimax problem, with minimization over a single scalar parameter (even for a multistock market). This interesting effect follows from the result of Chapter 6 for the optimal compression problem. Using this effect, the original maximin problem is solved explicitly; the optimal strategy is derived explicitly via solution of a linear parabolic equation.