764 research outputs found
The scale functions kit for first passage problems of spectrally negative Levy processes, and applications to the optimization of dividends
First passage problems for spectrally negative L\'evy processes with possible
absorbtion or/and reflection at boundaries have been widely applied in
mathematical finance, risk, queueing, and inventory/storage theory.
Historically, such problems were tackled by taking Laplace transform of the
associated Kolmogorov integro-differential equations involving the generator
operator. In the last years there appeared an alternative approach based on the
solution of two fundamental "two-sided exit" problems from an interval (TSE). A
spectrally one-sided process will exit smoothly on one side on an interval, and
the solution is simply expressed in terms of a "scale function" (Bertoin
1997). The non-smooth two-sided exit (or ruin) problem suggests introducing a
second scale function (Avram, Kyprianou and Pistorius 2004).
Since many other problems can be reduced to TSE, researchers produced in the
last years a kit of formulas expressed in terms of the " alphabet" for a
great variety of first passage problems. We collect here our favorite recipes
from this kit, including a recent one (94) which generalizes the classic De
Finetti dividend problem. One interesting use of the kit is for recognizing
relationships between apparently unrelated problems -- see Lemma 3. Last but
not least, it turned out recently that once the classic are replaced with
appropriate generalizations, the classic formulas for (absorbed/ reflected)
L\'evy processes continue to hold for:
a) spectrally negative Markov additive processes (Ivanovs and Palmowski
2012),
b) spectrally negative L\'evy processes with Poissonian Parisian absorbtion
or/and reflection (Avram, Perez and Yamazaki 2017, Avram Zhou 2017), or with
Omega killing (Li and Palmowski 2017)
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
Overshoots and undershoots of L\'{e}vy processes
We obtain a new fluctuation identity for a general L\'{e}vy process giving a
quintuple law describing the time of first passage, the time of the last
maximum before first passage, the overshoot, the undershoot and the undershoot
of the last maximum. With the help of this identity, we revisit the results of
Kl\"{u}ppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766--1801]
concerning asymptotic overshoot distribution of a particular class of L\'{e}vy
processes with semi-heavy tails and refine some of their main conclusions. In
particular, we explain how different types of first passage contribute to the
form of the asymptotic overshoot distribution established in the aforementioned
paper. Applications in insurance mathematics are noted with emphasis on the
case that the underlying L\'{e}vy process is spectrally one sided.Comment: Published at http://dx.doi.org/10.1214/105051605000000647 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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