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

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 do not provide any information depicting the optimal control. The diffusion term of the state equation is 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 performed to obtain the optimal uncertainty measure, among which the regularities of the state processes, variational processes, and adjoint processes (in the transposition sense) are carefully characterized. Malliavin calculus is applied to pave the way for Lebesgue differentiation theorem to deduce the pointwise robust optimality conditions.