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Stochastic Systems: Modeling, Identification and Optimization, I pp 22–38Cite as

Entrance-exit distributions for Markov additive processes

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Part of the book series: Mathematical Programming Studies ((MATHPROGRAMM,volume 5))

Abstract

Let X be a standard Markov process, and let S be a perfectly additive increasing process with conditionally independent increments given the paths of X. Then, (X, S) is a Markov additive process. Let C be the random time change associated with S, and put Z −t =X(C t -), Z + t =X(C t ), R − t =t-S(C t -), R + t =S(C t )-t. When the state space of X is finite, Getoor [5] has recently obtained the joint distribution of these variables in terms of a triple Laplace transform. Here, the same is obtained explicitly by using renewal theoretic arguments along with the results on Lévy systems of (X, S) given in Çinlar [4]. These results are useful in reliability theory and in the boundary theory of Markov processes.

Research supported by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No. AFOSR-74-2733. The United States Government is authorized to reproduce and distribute reprints for governmental purposes.

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Authors and Affiliations

  1. Northwestern University, Evanston, Ill., U.S.A.

    Erhan Çinlar

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  1. Erhan Çinlar
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Roger J.- B. Wets

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© 1976 The Mathematical Programming Society

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Çinlar, E. (1976). Entrance-exit distributions for Markov additive processes. In: Wets, R.J.B. (eds) Stochastic Systems: Modeling, Identification and Optimization, I. Mathematical Programming Studies, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120761

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  • DOI: https://doi.org/10.1007/BFb0120761

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00783-5

  • Online ISBN: 978-3-642-00784-2

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