Stochastic Control Problems for Systems Driven by Normal Martingales

In this paper we study a class of stochastic control problems in which the control of the jump size is essential. Such a model is a generalized version for various applied problems ranging from optimal reinsurance selections for general insurance models to queueing theory. The main novel point of such a control problem is that by changing the jump size of the system, one essentially changes the type of the driving martingale. Such a feature does not seem to have been investigated in any existing stochastic control literature. We shall first provide a rigorous theoretical foundation for the control problem by establishing an existence result for the multidimensional structure equation on a Wiener-Poisson space, given an arbitrary bounded jump size control process; and by providing an auxiliary counterexample showing the non-uniqueness for such solutions. Based on these theoretical results we then formulate the control problem and prove the Bellman Principle, and derive the corresponding Hamilton-Jacobi-Bellman (HJB) equation, which in this case is a mixed second-order partial differential/difference equation. Finally we prove a uniqueness result for the viscosity solution of such an HJB equation.