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This thesis contains six papers on the topics of optimal stopping and stochastic games. 

Paper I extends the classical Bayesian sequential testing and detection problems for a Brownian motion to higher dimensions. We demonstrate unilateral concavity of the cost function and present its structural properties through various examples.

Paper II studies the problem of sequentially testing two composite hypotheses concerning an unknown parameter within the exponential family, incorporating observation costs within the Bayesian setting. In a Markovian framework, we show that the value function is concave, and non-decreasing in time under certain assumptions, consequently leading to the monotonicity of the stopping boundaries.

Paper III formulates an optimal stopping problem involving an unknown state that influences the diffusion process drift, the payoff functions, and the distribution of the time horizon. By performing a measure change, we reformulate it into a two-dimensional stopping problem with full information. We further provide several examples where explicit solutions are possible.

Paper IV introduces some non-linear, non-local parabolic operators related to a tug-of-war game where the random waiting time is coupled with space. Following that, we state and prove the asymptotic mean value formulas of the fractional heat operator and the aforementioned operators, and discuss their probabilistic interpretations. 

Paper V considers a Dynkin game with consolation where the players act under asymmetric and incomplete information. We prove a verification result that allows us to identify a Nash equilibrium. Building upon this, we examine certain classes of problems where the equilibrium value functions and strategies can be constructed.

Paper VI addresses the Bayesian sequential estimation problem of an unknown parameter within the exponential family, considering observations with associated costs. We offer sufficient conditions for space monotonicity of the value function, and explore their consequential impacts on the structural attributes of continuation and stopping regions.