Abstract
In this chapter, we will deal with zero-sum two-player time-homogeneous stochastic differential games and viscosity solutions of the Isaacs equations arising from such games, via the dynamic programming principle.In Sect. 4.1, we are concerned with basic concepts and definitions and we introduce stochastic differential games, referring to (Controlled MarkovProcesses and viscosity solutions, 2nd edn. Springer, New York 2006), XI. Then, using a semi-discretization argument, we study the DPP for lower- and upper-value functions in Sect. 4.2. In Sect. 4.3, we will consider the Isaacs equations, via semigroups related to DPP. In Sect. 4.4, we consider a link between stochastic controls and differential games via risk sensitive controls.
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