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

Asmussen, Søren; Kella, Offer

doi:10.1239/aap/1013540169

Cited by 98 publications

(143 citation statements)

References 23 publications

(29 reference statements)

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“…It is the contribution of this appendix to show that this approach does not work, and to show how this can be resolved. In particular, we provide answers to the questions raised in Section 4 of Asmussen and Kella (2000) in the continuous-time Markov-additive context. Proposition 8.1 has implications for the locations of the singularities of D −− (α, β) in H + .…”

Section: Appendix: the Spectral Methods For The Matricesmentioning

confidence: 99%

“…A spectral analysis reveals the connection with this fixed-point equation, as detailed in the Appendix. In fact, the appendix outlines how a spectral analysis can also be used to find K α ∼∼ numerically, thereby complementing the discussion in Section 4 of Asmussen and Kella (2000).…”

Section: The Distribution Of (X F X I)mentioning

confidence: 99%

“…This formula is Eq. 4.1 of Asmussen and Kella (2000), who interpret u − in terms of local times. The following observation, however, is new.…”

Section: Proposition 42 Suppose That πmentioning

confidence: 99%

“…However, inspired by the work of Kennedy (1994), we are able to give appealing interpretations of all our results in terms of a last-passage process. Our approach to Markov-additive processes is essentially different from the occupation method developed in , the martingale method of Asmussen and Kella (2000), and the rate-conservation method of Miyazawa (2004).…”

Section: Introductionmentioning

confidence: 99%

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Extremes of Markov-additive Processes with One-sided Jumps, with Queueing Applications

Dieker¹,

Mandjes

2009

Methodol Comput Appl Probab

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Through Laplace transforms, we study the extremes of a continuoustime Markov-additive process with one-sided jumps and a finite-state background Markovian state-space, jointly with the epoch at which the extreme is 'attained'. For this, we investigate discrete-time Markov-additive processes and use an embedding to relate these to the continuous-time setting. The resulting Laplace transforms are given in terms of two matrices, which can be determined either through solving a nonlinear matrix equation or through a spectral method. Our results on extremes are first applied to determine the steady-state buffer-content distribution of several single-station queueing systems. We show that our framework comprises many models dealt with earlier, but, importantly, it also enables us to derive various new results. At the same time, our setup offers interesting insights into the connections between the approaches developed so far, including matrix-analytic techniques, martingale methods, the rate-conservation approach, and the occupation-measure method. We also study networks of fluid queues, and show how the results on single queues can be used to find the Laplace transform of the steady-state buffer-content vector; it has a matrix quasi-product form. Fluid-driven priority systems also have this property.

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Section: Appendix: the Spectral Methods For The Matricesmentioning

confidence: 99%

Section: The Distribution Of (X F X I)mentioning

confidence: 99%

“…This formula is Eq. 4.1 of Asmussen and Kella (2000), who interpret u − in terms of local times. The following observation, however, is new.…”

Section: Proposition 42 Suppose That πmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Extremes of Markov-additive Processes with One-sided Jumps, with Queueing Applications

Dieker¹,

Mandjes

2009

Methodol Comput Appl Probab

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“…If the claim is of size v then the process starts to decrease at rate 1 for v time units and then increases again at rate c; that is, in the fluid model, the downwards jumps are (pictorially speaking) replaced by segments with slope −1. This description was also used by Asmussen and Kella (2000) and Kella et al (2003). In Figure 1(b), we illustrate the fluid representation of R(t).…”

Section: (T) and Its Fluid Representation R (T)mentioning

confidence: 99%

On the expected time to ruin and the expected dividends when dividends are paid while the surplus is above a constant barrier

Frostig¹

2005

J. Appl. Probab.

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We study the expected time to ruin in a risk process in which dividends are paid when the surplus is above the barrier. We consider the case in which the dividend rate is smaller than the premium rate. We obtain results for the classical compound Poisson risk process with phase-type claim size. When the ruin probability is 1, we derive the expected time to ruin and the expected dividends paid. When the ruin probability is less than 1, these quantities are derived conditioning on the event that ruin occurs.

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L évy Processes

Kella¹

2007

Encyclopedia of Statistics in Quality and Reliability

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This entry contains an overview of Lévy processes and their multidimensional extensions, which are processes having stationary, independent increments that are continuous in probability thereby having versions with almost surely right continuous sample paths. Special cases are Poisson processes, Brownian motions and compound Poisson processes. Infinite divisibility, which is closely related to Lévy processes, is defined and discussed as well as the reflected version of these processes and associated martingales, both in the one‐ and multidimensional cases. Markov additive processes, which are extensions of Lévy process, as well as their reflected versions and associated martingales are introduced as well.

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

Cited by 98 publications

References 23 publications

Extremes of Markov-additive Processes with One-sided Jumps, with Queueing Applications

Extremes of Markov-additive Processes with One-sided Jumps, with Queueing Applications

On the expected time to ruin and the expected dividends when dividends are paid while the surplus is above a constant barrier

L évy Processes

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