Optimal inventory control with jump diffusion and nonlinear dynamics in the demand

In this paper, we consider an inventory control problem with a nonlinear evolution and a jump-diffusion demand model. This work extends the earlier inventory model proposed by Benkherouf and Johnson [Math. Methods Oper. Res., 76 (2012), pp. 377–393] by including a general jump process. However, as those authors note, their techniques are not applicable to models with demand driven by jump-diffusion processes with drift. Therefore, the combination of diffusion and general compound Poisson demands is not completely solved. From the dynamic programming principle, we transform the problem into a set of quasi-variational inequalities (Q.V.I.). The difficulty arises when solving the Q.V.I. because the second derivative and integration term appear in the same inequality. Our technique is to construct a set of coupled auxiliary functions. Then, an analytical study of the Q.V.I. implies the existence and uniqueness of an (s, S) policy.