Abstract
We study the existence and uniqueness of a solution for the multivalued stochastic differential equation with delay (the multivalued term is of subdifferential type):Specify that in this case the coefficients at time t depends also on previous values of X(t) 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 .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.
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