Reflecting or sticky Markov processes with Lévy generators as the limit of storage processes
The author considers the approximation to reflecting or sticky diffusions or more generally Markov processes with Lévy generators having nondegenerate diffusion coefficients of a sequence of storage processes of the following type, \[ X(t)= X(0)- \int_ 0^ t r(X(s)) ds+ \int_ 0^ t\;\int_{u>0} uN(ds, du)-\xi(t), \] where \(X(t)\) takes values in \([0,K]\) (possibly \(K=\infty\)), \(N(ds,dx)\) is an integer-valued random measure with compensator \(ds d\nu (u\mid X(s))\) and \(\xi(t)\) is increasing: \[ \xi(0)=0, \qquad \xi(t)= \int_ 0^ t 1_{\{K\}} (X(s)) d\xi(s), \] under suitable conditions imposed on \(r\) and \(d\nu (u\mid x)\). The author claims the general limit theorems for semimartingles cannot be directly applied owing to the special conditions posed in his approach. Some examples satisfying these conditions are given, including \textit{J. M. Harrison} and \textit{A. J. Lemoine's} model [J. Appl. Probab. 18, 216-226 (1981; Zbl 0453.60072)].
scientific article; zbMATH DE number 672065
scientific article