Journal of Applied Mathematics and Stochastic Analysis
Volume 7 (1994), Issue 3, Pages 457-464
doi:10.1155/S1048953394000365
First excess levels of vector processes
Department of Applied Mathematics, Florida Institute of Technology, Melbourne 32901, FL, USA
Received 1 April 1994; Revised 1 August 1994
Copyright © 1994 Jewgeni H. Dshalalow. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
This paper analyzes the behavior of a point process marked by a two-dimensional renewal process with dependent components about some fixed (two-dimensional) level. The compound process evolves until one of its marks hits (i.e.
reaches or exceeds) its associated level for the first time. The author targets a
joint transformation of the first excess level, first passage time, and the index of
the point process which labels the first passage time. The cases when both marks
are either discrete or continuous or mixed are treated. For each of them, an explicit and compact formula is derived. Various applications to stochastic models are
discussed.