Optimal choices: mean field games with controlled jumps and optimality in a stochastic volatility model

Decision making in continuous time under random influences is the leitmotif of this work. In the first part a family of mean field games with a state variable evolving as a jump-diffusion process is studied. Under fairly general conditions, the existence of a solution in a relaxed version of these games is established and conditions under which the optimal strategies are in fact Markovian are given. The proofs rely upon the notions of relaxed controls and martingale problems. Mean field games represent the limit, as the number of players tends to infinity, of nonzero-sum stochastic differential games. Under the assumption that the former admit a regular Markovian solution, an approximate Nash equilibrium for the corresponding n-player games is constructed, and the rate of convergence is provided. Finally, the general theory is applied to a simple illiquid inter-bank market model, where the banks can adjust their reserves only at the jump times of some given Poisson processes with a common constant intensity, and some numerical results are provided. In the second part a stochastic optimization problem is presented. Here the evolution of the state is modeled as in the Heston model, but with a further multiplicative control input in the volatility term. The main objective is to consider the possible role of an external actor, whose exogenous contribution is summarised in the control itself. The solvability of the Hamilton-Jacobi-Bellman equation associated to this optimal control problem is discussed.