Dimension reduction and mutual fund theorem in maximin setting for bond market

Abstract

We study optimal investment problem for a continuous time stochastic market model. 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. To cover fixed income management problems, we assume that the number of risky assets can be larger than the number of driving Brownian motion. 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 Mutual Fund Theorem holds for this setting. We found also that a saddle point exists and can be found via minimization over a single scalar parameter.