Extremal shift rule for continuous-time stochastic games

The talk is based on feedback control theory developed by Krasovskii and Subbotin. The talk concerns the problem of control design for the continuous time stochastic system governed by two players with the opposite purposes. The control with guide strategy is constructed. The motion of the guide is governed by a different stochastic system. The scheme of control with guide and with the stochastic guide can be realized for the differential game using the proposed approach. Additionally we consider the case when the genuine system is governed by differential equation whilst the guide is governed by the Markov chain. This lead tp the approximation of the value function of the differential game by the system of ODEs.