Markovian randomized equilibria in games of stopping

The thesis introduces two games of stopping based on a common stochastic model. For both games, equilibria in pure strategies are introduced. Then, Markovian randomized strategies are defined, discussed with respect to their properties and classified as a mathematical concept. Finally, we extend the equilibrium definitions of pure strategies to Markovian randomized strategies. This is followed by the three scientific contributions of the work.

Chapter 3 contains a proof of the existence of Markov-perfect equilibria in general nonzero-sum Dynkin games of war-of-attrition type, driven by a 1-dimensional Ito diffusion. This establishes Markovian randomized stopping times as the canonical type of equilibrium in the game. Another main contribution lies in the development of topologies for Markovian randomized stopping times on the basis of which the existence of equilibria can be proved by means of fixed point and general compactness arguments. 

In Chapter 4 we study a version of the equilibrium problem with randomized strategies that arises from the problem of optimally stopping a 1-dimensional Ito diffusion, subject to an expectation constraint on the time until stopping. We derive a verification theorem and necessary conditions for equilibria, which together basically characterize all equilibria. Furthermore, additional structural properties of equilibria are obtained to feed a possible guess-and-verify approach. It turns out that in order to determine the reward functional by the previously derived properties, we also need differentiability in order to have a sufficient number of conditions.

The required differentiability result is provided in Chapter 5. In addition, we conduct a full investigation of the continuity and smoothness of reward functionals associated with Markovian randomized stopping times. These properties can be used to apply a common approach to derive analytic expressions
for the reward functional.

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