Optimal control of stochastic delayed systems with jumps

Overview & Synopsis

We consider the optimal control of stochastic delayed systems with jumps, in which both the state and controls can depend on the past history of the system, for a value function which depends on the initial path of the process. We derive the Hamilton-Jacobi-Bellman equation and the associated verification theorem and prove a necessary and a sufficient maximum principles for such problems. Explicit solutions are obtained for a mean-variance portfolio problem and for the optimal consumption and portfolio case in presence of a financial market with delay and jumps.