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Theses

Jeux différentiels stochastiques à information incomplète

Abstract : The objective of this thesis is the study of stochastic differential games with incomplete information. We consider a game with two opponent players who control a diffusion in order to minimize, respectively maximize a certain payoff. To model the information incompleteness we will follow the famous ansatz 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 randomly. The information is then transmitted to one player while the second one only knows the respective probabilities for each state. In this thesis we establish a dual representation for stochastic differential games with incomplete information. Therein we make a vast use of the theory of backward stochastic differential equations (BSDEs), which turns out to be an indispensable tool in this study. Moreover we show how under some restrictions that this representation allows to construct optimal strategies for the informed player.Morover we give - using the dual representation - a strikingly simple proof for semiconvexity of the value function of differential games with incomplete information. Another part of this thesis is devoted to numerical schemes for stochastic differential games with incomplete information. In the last part we investigate continuous time optimal stopping games, so called Dynkin games, with information incompleteness. We show that these games have a value and a unique characterization by a fully non-linear variational PDE for which we provide a comparison principle. Also we establish a dual representation for Dynkin games with incomplete information.
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Christine Grün. Jeux différentiels stochastiques à information incomplète. Mathématiques générales [math.GM]. Université de Bretagne occidentale - Brest, 2012. Français. ⟨NNT : 2012BRES0017⟩. ⟨tel-00802378⟩

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