Multivalued stochastic delay differential equations and related stochastic control problems

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):

     dX(t) + aφ (X(t))dt ∍ b(t,X(t), Y(t), Z(t)) dt 

⎨                                  +σ (t, X (t), Y (t), Z (t)) dW (t), t ∈ (s, T)

      X(t) = ξ (t - s), t ∈ [s - δ, s].

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 Dom(φ).

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 nally we prove that the value function is a viscosity solution for a suitable Hamilton-Jacobi-Bellman type equation.

Keywords: Multivalued SDE with delay, dynamic programming principle, Hamilton-Jacobi-Bellman equation, viscosity solutions