Finite Moments for Inventory Processes

Abstract

We study a continuous time inventory process that is a reflection mapping of a semimartingale netput process. Inventory processes of this type include the workload process in queues, dam and storage processes (with perhaps pure jump Levy input), as well as processes arising in fluid models. We establish sufficient conditions on the netput ensuring that the steady-state inventory has finite moments of order $k \geq 1$, and derive explicit bounds for these moments. The sufficient conditions require that the netput have a negative (local) drift and that the (conditional) $(k + 1)$th moment of its increments be bounded.