971 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)
Optimal periodic dividend strategies for spectrally positive L\'evy risk processes with fixed transaction costs
We consider the general class of spectrally positive L\'evy risk processes,
which are appropriate for businesses with continuous expenses and lump sum
gains whose timing and sizes are stochastic. Motivated by the fact that
dividends cannot be paid at any time in real life, we study
dividend strategies whereby dividend decisions are made according to a separate
arrival process.
In this paper, we investigate the impact of fixed transaction costs on the
optimal periodic dividend strategy, and show that a periodic
strategy is optimal when decision times arrive according to an independent
Poisson process. Such a strategy leads to lump sum dividends that bring the
surplus back to as long as it is no less than at a dividend
decision time. The expected present value of dividends (net of transaction
costs) is provided explicitly with the help of scale functions. Results are
illustrated.Comment: Accepted for publication in Insurance: Mathematics and Economic
A generalized penalty function in Sparre Andersen risk models with surplus-dependent premium
In a general Sparre Andersen risk model with surplus-dependent premium income, the generalization of Gerber-Shiu function proposed by Cheung et al. (2010a) is studied. A general expression for such Gerber-Shiu function is derived, and it is shown that its determination reduces to the evaluation of a transition function which is independent of the penalty function. Properties of and explicit expressions for such a transition function are derived when the surplus process is subject to (i) constant premium; (ii) a threshold dividend strategy; or (iii) credit interest. Extension of the approach is discussed for an absolute ruin model with debit interest. © 2011 Elsevier B.V.postprin
- …