Relationship between maximum principle and dynamic programming principle for recursive optimal control problem of stochastic evolution equations

This paper aims to study the relationship between the maximum principle and the dynamic programming principle for recursive optimal control problem of stochastic evolution equations, where the control domain is not necessarily convex and the value function may be nonsmooth. By making use of the notion of conditionally expected operator-valued backward stochastic integral equations, we establish a connection between the first and second-order adjoint processes in MP and the general derivatives of the value function. Under certain additional assumptions, the value function is shown to be $C^{1,1}$-regular. Furthermore, we discuss the smooth case and present several applications of our results.