Finite moments for inventory processes
Let \(\{Z(t),\;t \geq 0\}\) be a continuous time inventory process which is a reflection mapping of a netput process \(\{X(t), t \geq 0\}\) with negative drift and \(X(0) = 0\). In other words \(Z(t) = X(t) + L(t)\), where \(L(t) = \sup_{0 \leq s \leq t} X^ -(s)\) with \(X^ -(s) = - \min (0, X(s))\). Processes of this type include workload process in queues and storage level in dam problems. The authors establish sufficient conditions on the netput which ensures the steady state inventory to have finite moments of order \(k \geq 1\) and obtain explicit bounds for these moments. The results are presented in detail, with examples, for the case when the netput has a bounded variation and are extended to the semi- martingale case.
scientific article; zbMATH DE number 681401