Stochastic differential games with incomplete information.

Summary The objective of this thesis is to study stochastic differential games with incomplete information. We consider a game with two opposing players who control a diffusion in order to minimize, respectively maximize a specific payoff. To model the incompleteness of information, we follow the famous approach of Aumann and Maschler. We assume that there are different states of nature in which the game can take place. Before the game starts, the state is chosen at random. In this thesis we establish a dual representation for differential stochastic games with incomplete information. Here, we make extensive use of the theory of stochastic backward differential equations (SDEs), which proves to be an indispensable tool in this study. Furthermore, we show how, under certain restrictions, this representation allows the construction of optimal strategies for the informed player. Then, using the dual representation, we give a particularly simple proof of the semiconvexity of the value function of differential games with incomplete information. Another part of the thesis is devoted to numerical schemes for stochastic differential games with incomplete information. In the last part we study optimal stopping games in continuous time, called Dynkin games, with incomplete information. We also establish a dual representation, which is used to determine optimal strategies for the informed player in this case.