Combined Singular and Impulse Control for Jump Diffusions

In this work we present and investigate the combined singular and impulse control problem for jump diffusions. Such problems frequently arise in finance, for instance, when both fixed and proportional transaction costs are considered. A verification theorem for the generalised combined singular and impulse control is formulated and proved. The verification theorem provides sufficient conditions for the existence of both the value function and optimal combined controls. An example is presented to illustrate the application of the theory.