Potential Measures of One-Sided Markov Additive Processes with Reflecting and Terminating Barriers

Consider a one-sided Markov additive process with an upper and a lower barrier, where each can be either reflecting or terminating. For both defective and nondefective processes, and all possible scenarios, we identify the corresponding potential measures, which help to generalize a number of results for one-sided Lévy processes. The resulting rather neat formulae have various applications in risk and queueing theories, and, in particular, they lead to quasistationary distributions of the corresponding processes.

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Potential Measures of One-Sided Markov Additive Processes with Reflecting and Terminating Barriers

Journal of Applied Probability ◽

10.1239/jap/1421763333 ◽

2014 ◽

Vol 51 (4) ◽

pp. 1154-1170 ◽

Cited By ~ 5

Author(s):

Jevgenijs Ivanovs

Keyword(s):

Lévy Processes ◽

Levy Processes ◽

Additive Process ◽

Markov Additive Process ◽

Additive Processes ◽

Quasistationary Distributions ◽

Markov Additive Processes

Consider a one-sided Markov additive process with an upper and a lower barrier, where each can be either reflecting or terminating. For both defective and nondefective processes, and all possible scenarios, we identify the corresponding potential measures, which help to generalize a number of results for one-sided Lévy processes. The resulting rather neat formulae have various applications in risk and queueing theories, and, in particular, they lead to quasistationary distributions of the corresponding processes.

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A multi-dimensional martingale for Markov additive processes and its applications

Advances in Applied Probability ◽

10.1017/s0001867800009988 ◽

2000 ◽

Vol 32 (02) ◽

pp. 376-393 ◽

Cited By ~ 12

Author(s):

Søren Asmussen ◽

Offer Kella

Keyword(s):

Brownian Motion ◽

Lévy Processes ◽

Levy Processes ◽

One Dimensional ◽

Additive Processes ◽

Motion Models ◽

And Brownian Motion ◽

Storage Processes ◽

Reflecting Barriers ◽

Markov Additive Processes

We establish new multidimensional martingales for Markov additive processes and certain modifications of such processes (e.g., such processes with reflecting barriers). These results generalize corresponding one-dimensional martingale results for Lévy processes. This martingale is then applied to various storage processes, queues and Brownian motion models.

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A multi-dimensional martingale for Markov additive processes and its applications

Advances in Applied Probability ◽

10.1239/aap/1013540169 ◽

2000 ◽

Vol 32 (2) ◽

pp. 376-393 ◽

Cited By ~ 57

Author(s):

Søren Asmussen ◽

Offer Kella

Keyword(s):

Brownian Motion ◽

Lévy Processes ◽

Levy Processes ◽

One Dimensional ◽

Additive Processes ◽

Motion Models ◽

And Brownian Motion ◽

Storage Processes ◽

Reflecting Barriers ◽

Markov Additive Processes

We establish new multidimensional martingales for Markov additive processes and certain modifications of such processes (e.g., such processes with reflecting barriers). These results generalize corresponding one-dimensional martingale results for Lévy processes. This martingale is then applied to various storage processes, queues and Brownian motion models.

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TRAFFIC GENERATED BY A SEMI-MARKOV ADDITIVE PROCESS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964810000227 ◽

2010 ◽

Vol 25 (1) ◽

pp. 21-27 ◽

Cited By ~ 1

Author(s):

Joke Blom ◽

Michel Mandjes

Keyword(s):

Positive Solution ◽

Linear Equations ◽

Levy Processes ◽

System Of Linear Equations ◽

Unique Positive Solution ◽

Exponential Distributions ◽

Background Process ◽

Additive Process ◽

Markov Additive Process ◽

Sojourn Times

We consider a semi-Markov additive process A(·)—that is, a Markov additive process for which the sojourn times in the various states have general (rather than exponential) distributions. Letting the Lévy processes Xi(·), which describe the evolution of A(·) while the background process is in state i, be increasing, it is shown how double transforms of the type $\vint_{0}^{\infty} e^{-qt}\, {\open E} \lsqb e^{-\alpha A(t)}\, {d}t \rsqb$ can be computed. It turns out that these follow, for given nonnegative α and q, from a system of linear equations, which has a unique positive solution. Several extensions are considered as well.

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Continuous-time Markov additive processes: Composition of large deviations principles and comparison between exponential rates of convergence

Journal of Applied Probability ◽

10.1239/jap/1011994182 ◽

2001 ◽

Vol 38 (4) ◽

pp. 917-931 ◽

Cited By ~ 4

Author(s):

Claudio Macci

Keyword(s):

Large Deviations ◽

Continuous Time ◽

Fluid Model ◽

Variational Formula ◽

Rates Of Convergence ◽

Rate Function ◽

Large Deviations Principle ◽

Additive Process ◽

Additive Processes ◽

Markov Additive Processes

We consider a continuous-time Markov additive process (Jt,St) with (Jt) an irreducible Markov chain on E = {1,…,s}; it is known that (St/t) satisfies the large deviations principle as t → ∞. In this paper we present a variational formula H for the rate function κ∗ and, in some sense, we have a composition of two large deviations principles. Moreover, under suitable hypotheses, we can consider two other continuous-time Markov additive processes derived from (Jt,St): the averaged parameters model (Jt,St(A)) and the fluid model (Jt,St(F)). Then some results of convergence are presented and the variational formula H can be employed to show that, in some sense, the convergences for (Jt,St(A)) and (Jt,St(F)) are faster than the corresponding convergences for (Jt,St).

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Continuous-time Markov additive processes: Composition of large deviations principles and comparison between exponential rates of convergence

Journal of Applied Probability ◽

10.1017/s0021900200019136 ◽

2001 ◽

Vol 38 (04) ◽

pp. 917-931 ◽

Cited By ~ 1

Author(s):

Claudio Macci

Keyword(s):

Large Deviations ◽

Continuous Time ◽

Fluid Model ◽

Variational Formula ◽

Rates Of Convergence ◽

Rate Function ◽

Large Deviations Principle ◽

Additive Process ◽

Additive Processes ◽

Markov Additive Processes

We consider a continuous-time Markov additive process (J t ,S t ) with (J t ) an irreducible Markov chain on E = {1,…,s}; it is known that (S t /t) satisfies the large deviations principle as t → ∞. In this paper we present a variational formula H for the rate function κ∗ and, in some sense, we have a composition of two large deviations principles. Moreover, under suitable hypotheses, we can consider two other continuous-time Markov additive processes derived from (J t ,S t ): the averaged parameters model (J t ,S t (A)) and the fluid model (J t ,S t (F)). Then some results of convergence are presented and the variational formula H can be employed to show that, in some sense, the convergences for (J t ,S t (A)) and (J t ,S t (F)) are faster than the corresponding convergences for (J t ,S t ).

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