  Optimal control and the associated second-order path-dependent
Hamilton-Jacobi-Bellman (PHJB) equation are studied for unbounded functional
stochastic evolution systems in Hilbert spaces. The notion of viscosity
solution without B-continuity is introduced in the sense of Crandall and Lions,
and is shown to coincide with the classical solutions and to satisfy a
stability property. The value functional is proved to be the unique continuous
viscosity solution to the associated PHJB equation, without assuming any
B-continuity on the coefficients. In particular, in the Markovian case, our
result provides a new theory of viscosity solutions to the
Hamilton-Jacobi-Bellman equation for optimal control of stochastic evolutionary
equations -- driven by a linear unbounded operator -- in a Hilbert space, and
removes the B-continuity assumption on the coefficients, which was initially
introduced for first-order equations by Crandall and Lions (see J. Func. Anal.
90 (1990), 237-283; 97 (1991), 417-465), and was subsequently used by Swiech
(Comm. Partial Differential Equations 19 (1994), 1999-2036) and Fabbri, Gozzi,
and Swiech (Probability Theory and Stochastic Modelling 82, 2017, Springer,
Berlin).
