Stochastic Cash Management Problem with Double Exponential Jump Diffusion Processes

In this paper, we investigate the effect of a sharp cash level fluctuation resulting from the inflow and outflow of a large amount of cash and how the cash balance is managed. We describe the cash level evolution as stochastic jump-diffusion process with double exponential distributed jump size, and formulate a cash management model for minimizing the sum of the transaction and holding-penalty costs. This model can be formulated as an impulse control model, and we derived the cost function under the assumption that a band-type policy exists. Moreover, we discuss the effect of such a fluctuation on the optimal policy though some numerical examples. Consequently, we show the cost function explicit and clarify that the size of sharp fluctuation has strong implications for the optimal policy.