Parabolic operators with unbounded coefficients with applications to stochastic optimal control games

The aim of this thesis is to improve some results on parabolic Cauchy problems with unbounded coefficients and their connection with stochastic optimal games. In the first part we summarize the recent results on parabolic operators with unbounded coefficients and on stochastic optimal control problem. In particular, in the matter of analyitic results, we recall the main exstence and uniqueness theorems for parabolic Cauchy problems with unbounded coefficients, the gradient estimates for the associated evolution operator, and its continuity and compactness properties. About the stochastic part, we briefly show the strong and weak formulation, which are the settings where the stochastic control problems are located, and we introduce the backward stochastic differential equations, which allow to connect a semilinear Cauchy problem with a class of stochastic control problem. In the second part we prove the existence and uniqueness of a mild solution to a semilinear parabolic Cauchy problem of Hamilton-Jacobi-Bellman (HJB) type. Moreover, we show that the solution to a Forward Backward Stochastic Differential Equation (FBSDE) can be expressed in terms of the solution to the HJB equation. Combining HJB equation and FBSDE, we show that, for a class of stochastic control problemin weak formulation, there exists an optimal control, and by means of the regularity of the solution to the HJB equation, we can identify the feedback law. The third part of the thesis is devoted to the study of a class of system of nonautonomous linear parabolic equations with unbounded coefficients, coupled both at first and zero order. We provide sufficient conditions which guarantee the existence and uniqueness of a classical solution to the Cauchy problem, and throughout this classical solution we define an evolution operator on the space of bounded and continuous functions. Further, we prove continuity properties of the evolution operator and that, under additional hypotheses, it is compact on the space of bounded and continuous functions. In the last chapter, we deal with a semilinear system of parabolic equations and its application to differential games. At first, we prove the existence of a mild solution to the system by an approximation argument. Throughout this mild solution, we show the existence of an adapted solution to a system of FBSDE which allows us to prove the existence of a Nash equilibrium for a class of differential games.