Optimal Impulse Control for Cash Balance Management in Jump-Diffusion Model

This paper is concerned with studying a problem that minimizes the total expected discounted cost over an infinite horizon for a cash management, where the cash fund follows jump-diffusion processes, holding-costs are assumed to be a general quadratic function of the cash level and there exist fixed and proportional transaction costs. Thanks to the variational inequality method in stochastic impulse control theory, we obtain the verification theorem, prove that the optimal control exists, and also gain its mathematical structures.