MEAN-FIELD STOCHASTIC CONTROL PROBLEMS UNDER SUBLINEAR EXPECTATION

Our work is devoted to the study of Pontryagin's stochastic maximum principle for a mean-field optimal control problem under Peng's G-expectation. The dynamics of the controlled state process is given by a stochastic differential equation driven by a G-Brownian motion, whose coefficients depend not only on the control and the controlled state process but also on its law under the G-expectation. Also the associated cost functional is of mean-field type. Under the assumption of a convex control state space we study the stochastic maximum principle, which gives a necessary optimality condition for control processes. Under additional convexity assumptions on the Hamiltonian it is shown that this necessary condition is also a sufficient one. Finally, as an application, we give an example of a control problem. The main difficulty which we have to overcome in our work consists in the differentiation of the G-expectation of parameterized random variables. As particularly delicate it turns out to handle with the G-expectation of a function of the controlled state process inside the running cost of the cost function. For this we have to study a measurable selection theorem for set-valued functions whose values are subsets of the representing set of probability measures for the G-expectation.