Sup-Inf/Inf-Sup Problem on Choice of a Probability Measure by Forward–Backward Stochastic Differential Equation Approach

Abstract

This article presents a problem on model uncertainties in stochastic control, in which an agent assumes a best case scenario on one risk and at the same time a worst case scenario on another risk. Particularly, the agent maximizes its view on a Brownian motion, simultaneously minimizing its view on another Brownian motion in choice of a probability measure. This selection method of a probability measure generalizes an approach to model uncertainties in which one considers the worst case scenarios for the views on Brownian motions, such as in the robust control. Specifically, we newly formulate and solve this problem based on a backward stochastic differential equation (BSDE) approach as a sup-inf (resp., inf-sup) optimal control problem on choice of a probability measure with the control domains dependent on stochastic processes. Concretely, we show that under certain conditions, the sup-inf and inf-sup problems are equivalent and these are solved by finding a solution of a BSDE with a stochastic Lipschitz driver. Then, we investigate two cases in which the optimal probability measure is explicitly obtained. The expression of the optimal probability measure includes signs of the diffusion terms of the value process, which are hard to determine in general. In these cases, we show two methods of determining the signs:the first one is by comparison theorems, and the second one is to predetermine the signs a priori and confirm them afterward by explicitly solving the corresponding equations.