25,229 research outputs found
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
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
Power identities for L\'evy risk models under taxation and capital injections
In this paper we study a spectrally negative L\'evy process which is
refracted at its running maximum and at the same time reflected from below at a
certain level. Such a process can for instance be used to model an insurance
surplus process subject to tax payments according to a loss-carry-forward
scheme together with the flow of minimal capital injections required to keep
the surplus process non-negative. We characterize the first passage time over
an arbitrary level and the cumulative amount of injected capital up to this
time by their joint Laplace transform, and show that it satisfies a simple
power relation to the case without refraction. It turns out that this identity
can also be extended to a certain type of refraction from below. The net
present value of tax collected before the cumulative injected capital exceeds a
certain amount is determined, and a numerical illustration is provided
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
One-dimensional reflected diffusions with two boundaries and an inverse first-hitting problem
We study an inverse first-hitting problem for a one-dimensional,
time-homogeneous diffusion reflected between two boundaries and
which starts from a random position Let be a given
threshold, such that and an assigned distribution
function. The problem consists of finding the distribution of such that
the first-hitting time of to has distribution This is a
generalization of the analogous problem for ordinary diffusions, i.e. without
reflecting, previously considered by the author
First passage problems for upwards skip-free random walks via the paradigm
We develop the theory of the and scale functions for right-continuous
(upwards skip-free) discrete-time discrete-space random walks, along the lines
of the analogue theory for spectrally negative L\'evy processes. Notably, we
introduce for the first time in this context the one and two-parameter scale
functions , which appear for example in the joint problem of deficit at ruin
and time of ruin, and in problems concerning the walk reflected at an upper
barrier. Comparisons are made between the various theories of scale functions
as one makes time and/or space continuous. The theory is shown to be fruitful
by providing a convenient unified framework for studying dividends-capital
injection problems under various objectives, for the so-called compound
binomial risk model of actuarial science.Comment: 27 page
Two-sided reflected Markov-modulated Brownian motion with applications to fluid queues and dividend payouts
In this paper we study a reflected Markov-modulated Brownian motion with a
two sided reflection in which the drift, diffusion coefficient and the two
boundaries are (jointly) modulated by a finite state space irreducible
continuous time Markov chain. The goal is to compute the stationary
distribution of this Markov process, which in addition to the complication of
having a stochastic boundary can also include jumps at state change epochs of
the underlying Markov chain because of the boundary changes. We give the
general theory and then specialize to the case where the underlying Markov
chain has two states. Moreover, motivated by an application of optimal dividend
strategies, we consider the case where the lower barrier is zero and the upper
barrier is subject to control. In this case we generalized earlier results from
the case of a reflected Brownian motion to the Markov modulated case.Comment: 22 pages, 1 figur
Power identities for Lévy risk models under taxation and capital injections
In this paper we study a spectrally negative Lévy process which is refracted at its running maximum and at the same time reflected from below at a certain level. Such a process can for instance be used to model an insurance surplus process subject to tax payments according to a loss-carry-forward scheme together with the flow of minimal capital injections required to keep the surplus process non-negative. We characterize the first passage time over an arbitrary level and the cumulative amount of injected capital up to this time by their joint Laplace transform, and show that it satisfies a simple power relation to the case without refraction, generalizing results by Albrecher and Hipp (2007) and Albrecher, Renaud and Zhou (2008). It turns out that this identity can also be extended to a certain type of refraction from below. The net present value of tax collected before the cumulative injected capital exceeds a certain amount is determined, and a numerical illustration is provided
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