Generalized Itô's formulae and their applications to optimal control problems with delay

Abstract

Stochastic optimal control problems have various applications in economics, mathematical finance, engineering and other fields. To provide a more realistic model, it is practical to add the delay effects in the problem setting. In Chapters 1 and 2, we will review some important examples of stochastic optimal control problems with delay effects in the existing literature. In particular, the general framework of mean field game and mean field type control problems, as well as their counterparts with delay effects are introduced. Their associated forward-backward partial differential equations (PDEs) and the master equations are also compared. To study the optimal control problems, various versions of Itô’s formulae play a crucial role, such as in deriving the Hamilton-Jacobi-Bellman equation or maximum principle. When the delay effects or the mean field effect are added, one requires the functional Itô’s formulae to study the problems. We will review a series of generalized Itô’s formulae appearing in the literature and how they can be applied to solve delay control problems in Chapter 2 as well. In Chapter 3, combining the functional Itô’s formula’s framework in [22] and the mean field Itô’s formula in [10], we present a new functional Itô’s formula for functions with the delay of mean field term. To explain the new functional Itô’s formula, we will also define the concept of vertical Gâteaux derivative and horizontal derivative, and introduce some assumptions. Some examples will be provided to demonstrate how to apply the formula. In Chapter 4, we will demonstrate how to apply this formula to dynamic programming in the setting of mean field type control problems.