Topics in Mean-Field Control and Games for Pure Jump Processes

Sammanfattning: This thesis is the collection of four papers addressing topics in stochastic optimal control, zero-sum games, backward stochastic differential equations, Pontryagin stochastic maximum principle and relaxed stochastic optimal control.In the first two papers, we establish existence of Markov chains of mean-field type, with countable state space and unbounded jump intensities. We further show existence of nearly-optimal controls and, using a Markov chain backward SDE approach, we derive conditions for existence of an optimal control and a saddle-point for a zero-sum differential game associated with risk-neutral and risk-sensitive payoff functionals of mean-field type, under dynamics driven by Markov chains of mean-field type. Our formulation of the control problems is of weak-type, where the dynamics are given in terms of a family of probability measures, under which the coordinate process is a pure jump process with controlled jump intensities.In the third paper, we characterize the optimal controls obtained in the first pa-per by deriving sufficient and necessary optimality conditions in terms of a stochastic maximum principle (SMP). Finally, within a completely different setup, in the fourth paper we establish existence of an optimal stochastic relaxed control for stochastic differential equations driven by a G-Brownian motion.