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In this article, a notion of viscosity solutions is introduced for second-order Hamilton–Jacobi–Bellman (HJB) equations with delays associated with optimal control problems for stochastic differential equations with delays. We identify the value functional of optimal control problem as the unique viscosity solution to the second-order HJB equation. [ABSTRACT FROM AUTHOR]

In this paper, a controlled stochastic delay heat equation with Neumann boundary-noise and boundary-control is considered. The existence and uniqueness of the mild solution for the associated Hamilton–Jacobi–Bellman equations are obtained by means of the backward stochastic differential equations, which is applied to the optimal control problem. [ABSTRACT FROM AUTHOR]

Abstract: By means of backward stochastic differential equations, the existence and uniqueness of the mild solution are obtained for the nonlinear Kolmogorov equations associated with stochastic delay evolution equations. Applications to optimal control are also given. [Copyright &y& Elsevier]