Multivalued Stochastic Delay Differential Equations and Related Stochastic Control Problems

We study the existence and uniqueness of a solution for the multivalued stochastic differential equation with delay (the multivalued term is of subdifferential type): \left\{\begin{array} [c]{r} dX(t)+\partial\varphi\left(X(t)\right) dt\ni b\left(t,X(t),Y(t),Z(t)\right) dt+\sigma\left(t,X(t),Y(t),Z(t)\right)dW(t), \medskip\\ t\in(s,T],\medskip\\ \multicolumn{1}{l}{X(t)=\xi\left(t-s\right) ,\;t\in\left[ s-\delta,s\right] .} \end{array} \right. Specify that in this case the coefficients at time $t$ depends also on previous values of $X\left(t\right)$ through $Y(t)$ and $Z(t)$. Also $X$ is constrained with the help of a bounded variation feedback law $K$ to stay in the convex set $\bar{\mathrm{Dom}\left(\varphi\right)}$. Afterwards we consider optimal control problems where the state $X$ is a solution of a controlled delay stochastic system as above. We establish the dynamic programming principle for the value function and finally we prove that the value function is a viscosity solution for a suitable Hamilton-Jacobi-Bellman type equation.