273 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)
The Class of Semi-Markov Accumulation Processes
In conjunction with the 15th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2017)International audienceIn this paper, we introduce a new accumulation process, the Semi-Markov Accumulation Process (SMAP). This class of processes extends the framework of continuous-time Markov Additive Processes (MAPs) by allowing the underlying environmental component to be a semi-Markov process instead of a Markov process. Next, we follow an analytic approach to derive a Master Equation formula of the Renewal type that describes the evolution of SMAPs in time. We show that under exponential holding times, a matrix exponential form analogous to the matrix exponent of a MAP is attained. Finally, we consider an application of our results where closed-form solutions are rather easy to achieve
Gerber-Shiu theory for discrete risk processes in a regime switching environment
In this paper we develop the Gerber-Shiu theory for the classic and dual discrete risk processes in a Markovian (regime switching) environment. In particular, by expressing the Gerber-Shiu function in terms of potential measures of an upward (downward) skip-free discrete-time and discrete-space Markov Additive Process (MAP), we derive closed form expressions for the Gerber-Shiu function in terms of the so-called (discrete) Wv and Zv scale matrices, which were introduced in [27]. We show that the discrete scale matrices allow for a unified approach for identifying the Gerber-Shiu function as well as the value function of the associated constant dividend barrier problems
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
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