Asymptotic irrelevance of initial conditions for Skorohod reflection mapping on the nonnegative orthant.

*(English)*Zbl 1248.90029The authors present deterministic reflection mappings and the associated Skorohod problem. Since they extend this framework to the case of stochastic processes, they examine the convergence of the reflected process.

Also, problems associated to the convergence of the translated and reflected process are studied. For the case of the translated reflected process, the renewal risk model is considered as an example. An extended theorem as the central result of the article is also given, which describes the convergence and the stationary properties of the reflected processes both with the properties of the translated reflected processes. The properties of the reflected Lévy and Markovian processes are also provided with respect to these results. The final part of the article is devoted to some extensions of the existing results related to the cases of non-constant reflection matrix and drift.

Also, problems associated to the convergence of the translated and reflected process are studied. For the case of the translated reflected process, the renewal risk model is considered as an example. An extended theorem as the central result of the article is also given, which describes the convergence and the stationary properties of the reflected processes both with the properties of the translated reflected processes. The properties of the reflected Lévy and Markovian processes are also provided with respect to these results. The final part of the article is devoted to some extensions of the existing results related to the cases of non-constant reflection matrix and drift.

Reviewer: Christos E. Kountzakis (Karlovassi)