  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
nonuniqueness 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.
