Reflected diffusions defined via the extended Skorokhod map
Kavita Ramanan, Carnegie Mellon University
Abstract
This work introduces the extended Skorokhod problem (ESP) and associated
extended Skorokhod map (ESM) that enable a pathwise construction of reflected
diffusions that are not necessarily semimartingales. Roughly speaking, given
the closure G of an open connected set in R J, a non-empty convex cone d(x) in
R J, specified at each point x on the boundary of G, and a cadlag trajectory
ψ taking values in R J, the ESM defines a constrained version φ of ψ
that takes values in G and is such that the increments of φ - ψ on any
interval [s,t] lie in the closed convex hull of the directions d(φ(u)), u ∈
(s,t]. General deterministic properties of the ESP are first established under
the only assumption that the graph of d(.) is closed. Next, for a class of
multi-dimensional ESPs on polyhedral domains, pathwise uniqueness and existence
of strong solutions to the associated stochastic differential equations is
established. In addition, it is also proved that these reflected diffusions are
semimartingales on [0,τ0], where τ0 is the time to hit the set of
points x on the boundary for which d(x) contains a line. One motivation for the
study of this class of reflected diffusions is that they arise as
approximations of queueing networks in heavy traffic that use the so-called
generalised processor sharing discipline.
Full text: PDF | PostScript
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona. You can even send a scanned jpeg or pdf of this copyright form to the managing editor ejpecpme@math.univ-nantes.fr. as an attached file.
If a paper has several authors, the corresponding author signs the copyright form on behalf of all the authors.