Optimal Control And Large Deviations Of Mean-field Stochastic Differential Systems

Mckean-Vlasov equation was introduced by Kac in1956. The theory of Mckean-Vlasov equation has been studied with great interest, since it accurately describes this kind of phenomenon in real world:the system exists interactions, thus the motion of a single individual not only rely on its own motion state, at the same time, also depends on the mean motion state of the whole system. Such models are often referred to as mean-field stochastic differential equations. Due to their wide applications in statistical physics, biology, financial engineering, social science, control theory, and so on, mean-field stochastic differential equations become a popular topic. The optimal control problems for mean-field stochastic differential equations are different from the optimal control problems for stochastic differential equations. In fact, they lead to the so-called time inconsistent control problems. That is, the Bellman optimality principle does not hold. Therefore, it is important to study the optimal control problems for mean-field stochastic differential equations. This paper mainly studies optimal control problems and large deviations for mean-field stochastic differential equations.The large deviation control problem is an important part of stochastic optimal con-trol. Thanks to duality method, the dual problem of the large deviation control problem is an ergodic risk sensitive control problem. It is known that HJB equation is effective to the classical risk sensitive control problems. In finite time horizon, we need to solve an HJB equation, in addition, we need to discuss an ergodic HJB equation in order to solve an ergodic risk sensitive control problem. However, the HJB equation is invalid for risk sensitive control problem of mean-field type. Therefore, it makes perfect sense to discuss large deviation control problems and risk sensitive control problems of mean-field type. With the help of an auxiliary process, the risk sensitive control problems of mean-field type turn to the classical risk sensitive control problems. Then, by means of results of HJB equations and Riccati equations, we effectively solve the risk sensitive control prob- lems of mean-field type; furthermore, we obtain the optimal strategies and optimal value functions of the primal large deviation control problems.Huang, Li and Yong (2012) introduced mixed linear-quadratic optimal control prob-lems, we extend them to mean-field stochastic differential equations case. Solving a mixed linear-quadratic optimal control problem for mean-field stochastic differential equations is equivalent to sequentially solve a two stage project linear-quadratic optimal control problem for mean-field stochastic differential equations and a deterministic optimal time problem.A new kind of mean-field stochastic differential equations and a new class of back-ward differential equations—mean-field backward stochastic differential equations were introduced by Buckdahn, Li and Peng (2009). We are interested in the large deviation and moderate deviation problems of these type processes with small random perturbation. First, thanks to a variational representation of functional for an finite dimensional Brown motion, we apply the weak convergence approach to establish the large deviation principle of mean-field stochastic differential equations with small random perturbation. Second, we extend the moderate deviation principle of classical stochastic differential equations to mean-field stochastic differential equations case. At last, we adopt the generalized contraction principle and the Delta method to derive the large deviation and moderate deviation principles of mean-field forward-backward stochastic differential equations with small random perturbation.