## Impulse Control Problems: Solution and Modeling

**Speaker: **Chao Zhu from University of Wisconsin-Milwaukee

**Date and Time:** 03/16/20, 4-5pm

**Room:** SH306

**Abstract: **This talk starts with an optimal inventory control problem using a long-term average criterion. In absence of ordering, the inventory process is modeled by a one-dimensional diffusion on some interval of (-∞, ∞) with general drift and diffusion coefficients and boundary points that are consistent with the notion that demands tend to reduce the inventory level. Orders instantaneously increase the inventory level and incur both positive fixed and level dependent costs. In addition, state-dependent holding/backorder costs are incurred continuously. Using average expected occupation and ordering measures and weak convergence arguments, weak conditions are given for the optimality of the (s_{*}, S_{*}) ordering policy in the general class of admissible policies. The second part of the talk is devoted to the construction of the probability measure on a single path space for an admissible intervention policy subject to an uncertain impulse mechanism. An important feature of this construction is that the intervention times are stopping times with respect to the natural filtration. In addition, when the intervention policy results in deterministic distributions for each impulse, the paths between interventions are independent and, moreover, if the same distribution is used for each impulse, then the cycles following the initial cycle are identically distributed. We also identify a class of impulse policies under which the resulting controlled process is Markov. The decision to use an (s, S) ordering policy in inventory management provides an example of an impulse policy for which the process is Markov and has i.i.d.~cycles so a benefit of the constructed model is that one is allowed to use classical renewal arguments.