306 research outputs found
Occupation densities in solving exit problems for Markov additive processes and their reflections
This paper solves exit problems for spectrally negative Markov additive
processes and their reflections. A so-called scale matrix, which is a
generalization of the scale function of a spectrally negative \levy process,
plays a central role in the study of exit problems. Existence of the scale
matrix was shown in Thm. 3 of Kyprianou and Palmowski (2008). We provide a
probabilistic construction of the scale matrix, and identify the transform. In
addition, we generalize to the MAP setting the relation between the scale
function and the excursion (height) measure. The main technique is based on the
occupation density formula and even in the context of fluctuations of
spectrally negative L\'{e}vy processes this idea seems to be new. Our
representation of the scale matrix W(x)=e^{-\Lambda x}\eL(x) in terms of nice
probabilistic objects opens up possibilities for further investigation of its
properties
First passage of a Markov additive process and generalized Jordan chains
In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions. This result provides us with a technique, which can be used to derive various further identities.Lévy processes, Fluctuation theory, Markov Additive Processes
On the record process of time-reversible spectrally-negative Markov additive processes
We study the record process of a spectrally-negative Markov additive process (MAP). Assuming time-reversibility, a number of key quantities can be given explicitly. It is shown how these key quantities can be used when analyzing the distribution of the all-time maximum attained by MAPs with negative drift, or, equivalently, the stationary workload distribution of the associated storage system; the focus is on Markov-modulated Brownian mo- tion, spectrally-negative and spectrally-positive MAPs. It is also argued how our results are of great help in the numerical analysis of systems in which the driving MAP is a superposition of multiple time-reversible MAPs
Markov-modulated Brownian motion with two reflecting barriers
We consider a Markov-modulated Brownian motion reflected to stay in a strip
[0,B]. The stationary distribution of this process is known to have a simple
form under some assumptions. We provide a short probabilistic argument leading
to this result and explaining its simplicity. Moreover, this argument allows
for generalizations including the distribution of the reflected process at an
independent exponentially distributed epoch. Our second contribution concerns
transient behavior of the reflected system. We identify the joint law of the
processes t,X(t),J(t) at inverse local times.Comment: 13 pages, 1 figur
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