10 research outputs found
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
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
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 time-reversible spectrally negative Markov additive processes
We study the first passage process of a spectrally negative Markov additive process (MAP). The focus is on the background Markov chain at the times of the first passage. This process is a Markov chain itself with a transition rate matrix ¿. Assuming time reversibility, we show that all the eigenvalues of ¿ are real, with algebraic and geometric multiplicities being the same, which allows us to identify the Jordan normal form of ¿. Furthermore, this fact simplifies the analysis of fluctuations of a MAP. We provide an illustrative example and show that our findings greatly reduce the computational efforts required to obtain ¿ in the time-reversible case
First passage of time-reversible spectrally negative Markov additive processes
We study the first passage process of a spectrally negative Markov additive process (MAP). The focus is on the background Markov chain at the times of the first passage. This process is a Markov chain itself with a transition rate matrix ¿. Assuming time reversibility, we show that all the eigenvalues of ¿ are real, with algebraic and geometric multiplicities being the same, which allows us to identify the Jordan normal form of ¿. Furthermore, this fact simplifies the analysis of fluctuations of a MAP. We provide an illustrative example and show that our findings greatly reduce the computational efforts required to obtain ¿ in the time-reversible case