51 research outputs found
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
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
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
A bivariate risk model with mutual deficit coverage
We consider a bivariate Cramer-Lundberg-type risk reserve process with the
special feature that each insurance company agrees to cover the deficit of the
other. It is assumed that the capital transfers between the companies are
instantaneous and incur a certain proportional cost, and that ruin occurs when
neither company can cover the deficit of the other. We study the survival
probability as a function of initial capitals and express its bivariate
transform through two univariate boundary transforms, where one of the initial
capitals is fixed at 0. We identify these boundary transforms in the case when
claims arriving at each company form two independent processes. The expressions
are in terms of Wiener-Hopf factors associated to two auxiliary compound
Poisson processes. The case of non-mutual (reinsurance) agreement is also
considered
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