3,587 research outputs found
Factorization identities for reflected processes, with applications
We derive factorization identities for a class of preemptive-resume queueing
systems, with batch arrivals and catastrophes that, whenever they occur,
eliminate multiple customers present in the system. These processes are quite
general, as they can be used to approximate Levy processes, diffusion
processes, and certain types of growth-collapse processes; thus, all of the
processes mentioned above also satisfy similar factorization identities. In the
Levy case, our identities simplify to both the well-known Wiener-Hopf
factorization, and another interesting factorization of reflected Levy
processes starting at an arbitrary initial state. We also show how the ideas
can be used to derive transforms for some well-known
state-dependent/inhomogeneous birth-death processes and diffusion processes
Meromorphic Levy processes and their fluctuation identities
The last couple of years has seen a remarkable number of new, explicit
examples of the Wiener-Hopf factorization for Levy processes where previously
there had been very few. We mention in particular the many cases of spectrally
negative Levy processes, hyper-exponential and generalized hyper-exponential
Levy processes, Lamperti-stable processes, Hypergeometric processes,
Beta-processes and Theta-processes. In this paper we introduce a new family of
Levy processes, which we call Meromorphic Levy processes, or just M-processes
for short, which overlaps with many of the aforementioned classes. A key
feature of the M-class is the identification of their Wiener-Hopf factors as
rational functions of infinite degree written in terms of poles and roots of
the Levy-Khintchin exponent, all of which appear on the imaginary axis of the
complex plane. The specific structure of the M-class Wiener-Hopf factorization
enables us to explicitly handle a comprehensive suite of fluctuation identities
that concern first passage problems for finite and infinite intervals for both
the process itself as well as the resulting process when it is reflected in its
infimum. Such identities are of fundamental interest given their repeated
occurrence in various fields of applied probability such as mathematical
finance, insurance risk theory and queuing theory.Comment: 12 figure
A note on Wiener-Hopf factorization for Markov Additive processes
We prove the Wiener-Hopf factorization for Markov Additive processes. We
derive also Spitzer-Rogozin theorem for this class of processes which serves
for obtaining Kendall's formula and Fristedt representation of the cumulant
matrix of the ladder epoch process. Finally, we also obtain the so-called
ballot theorem
First passage process of a Markov additive process, with applications to reflection problems
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. Importantly, our
result also provides us with a technique, which can be used to derive various
further identities. We then proceed to show how to compute the stationary
distribution associated with a one-sided reflected (at zero) MAP for both the
spectrally positive and spectrally negative cases as well as for the two sided
reflected Markov-modulated Brownian motion; these results can be interpreted in
terms of queues with MAP input.Comment: 16 page
A new approach to fluctuations of reflected L\'{e}vy processes
We present a new approach to fluctuation identities for reflected L\'{e}vy
processes with one-sided jumps. This approach is based on a number of easy to
understand observations and does not involve excursion theory or It\^{o}
calculus. It also leads to more general results.Comment: 6 page
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