Properties for transposition solutions to operator-valued BSEEs, and applications to robust second order necessary conditions for controlled SEEs

Abstract

This article is concerned with the second order necessary conditions for the stochastic optimal control problem of stochastic evolution equation with model uncertainty when the traditional Pontryagin-type maximum principle holds trivially and does not provide any information depicting the optimal control. The diffusion terms of the state equations are allowed to be control dependent with convex control constraints. Transposition method is adopted to deal with the adjoint operator-valued backward stochastic evolution equations, especially the correction terms. Besides, weak convergence arguments are used to obtain the optimal uncertainty reference measure, in which the regularities of the state processes, variational processes, and adjoint processes in the transposition sense are characterized. Malliavin calculus is applied to pave the way for differentiation theorem of Lebesgue type to deduce the pointwise robust optimality conditions.