Pontryagin-Type Stochastic Maximum Principle and Beyond

Abstract

The main purpose of this chapter is to derive first order necessary conditions, i.e., Pontryagin-type maximum principle for optimal controls of general nonlinear stochastic evolution equations in infinite dimensions, in which both the drift and the diffusion terms may contain the control variables, and the control regions are allowed to be nonconvex. In order to do this, quite different from the deterministic infinite dimensional setting and the stochastic finite dimensional case, people have to introduce a suitable operator-valued backward stochastic evolution equation, served as the second order adjoint equation. It is very difficult to prove the existence of solutions to this equation for the general case. Indeed, in the infinite dimensional setting, there exists no such a satisfactory stochastic integration/evolution equation theory (in the previous literatures) that can be employed to establish the well-posedness of such an equation.