Risk analysis for a stochastic cash manangement model with two type of customers
The authors study the risk analysis of a stochastic cash management model possessing two types of customers: many ``small'' customers, frequently depositing and withdrawing small amounts of money, and a few ``big'' ones causing big upward or downward jumps in the amount of cash held by the system. Accordingly, the total amount of cash in the system is modeled as a sum of two independent components, a Brownian motion with drift plus a compound Poisson process with (positive or negative) exponential jumps, such that the total drift of the cash flow is negative. Making use of some martingale arguments for stochastic storage processes with Lévy input due to \textit{O. Kella} and \textit{W. Whitt} [J. Appl. Probab. 29, 396-403 (1992; Zbl 0761.60065)], the authors derive explicit formulae for the Laplace transforms of some key system characteristics such as e.g. the time to bankruptcy, the maximum amount of cash before bankruptcy, or the expected discounted revenue generated by the system. Possible extensions to models with phase-type or hyperexponential jumps are also discussed.
scientific article; zbMATH DE number 1441570
scientific article