A remark on Skorokhod topologies for the Skorokhod reflection problem (Q2726258)

The reflection problem was defined by \textit{A. V. Skorokhod} [Theory Probab. Appl. 6, 264-274 (1961), translation from Teor. Veroyatn. Primen. 6, 287-298 (1961; Zbl 0215.53501); ibid. 7, 3-23 (1962), resp. ibid. 7, 5-25 (1962)]. It is very easy to state for a continuous process, the behaviour of jumps may lead to different definitions for the very same problem as soon as we deal with continuous processes. As a matter of fact, there are at least two different definitions of the Skorokhod reflection problem in that setting. The first is for a one-dimensional process with reflecting boundary being \(R\times\{0\}\). The other one is for multidimensional processes and a far more general class of domains, with additional stability properties under Skorokhod's \(J_{1}\)-topology. These definitions are called respectively definitions (C) and (S). The aim of this paper is to give a topological argument hinting that definition (S) is the proper generalization of the continuous Skorokhod reflection problem to discontinuous processes. It is proved that if the processes are considered reflecting on the half-line, then definition (S), and only this one, is stable under convergence in Skorokhod's \(M_{1}\)-topology.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00022].