It has been decades since the first paper that mean field problems were studied. More and more problems are considered or solved as new methods and new concepts have been developed. In this dissertation, we will present a series of results on (recursive) mean field stochastic optimal control problems. Comparing our results with those in the classical stochastic optimal control theory, there are following significant differences. First, the value function of a mean field optimal control problem is not Markovian any more, even when coefficient functions in the problem are deterministic. Second, the cost functional we considered is induced by a mean field backward stochastic differential equation. This leads to the value function to be random. Last but not the least, the backward stochastic differential equation we considered is of McKean-Vlasov form. The appearance of the distribution of its solution Y at time s leads to a new Hamiltion-Jacobi-Bellman equation. To overcome these difficulties, we first introduce an auxiliary problem associated with the original optimal control problem, so that we can better analyze the dependence of the value function V on the initial state . We also give a description of optimal control by a necessary condition, which is derived from the Hamiltion-Jacobi-Bellman equation. About this new HJB equation, we will prove the verification theorem and introduce the notion of viscosity solution.
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Doctor of Philosophy (Ph.D.)
College of Sciences
Length of Campus-only Access
Doctoral Dissertation (Open Access)
Yan, Wei, "Mean Field Optimal Control and Related Problems" (2020). Electronic Theses and Dissertations, 2020-. 316.