Jewgeni H. Dshalalow, "First excess levels of vector processes", International Journal of Stochastic Analysis, vol. 7, Article ID 762639, 8 pages, 1994. https://doi.org/10.1155/S1048953394000365
First excess levels of vector processes
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.
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