605 research outputs found
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
Dependent Risks and Ruin Probabilities in Insurance
Classical risk process models in insurance rely on independency. However, especially when modeling natural events, this assumption is very restrictive. This paper proposes a new approach to introducing dependency structures between events into the model and investigates its effects on a crucial parameter for insurance companies, the probability of ruin. Explicit formulas, numerical simulations and sensitivity results for dependence are established for different dependency models of first-order markovian type indicating that for various scenarios dependency considerably increases the probability of ruin
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)
Static hedging of Asian options under Lévy models: the comonotonicity approach.
In this paper we present a simple static super-hedging strategy for the payoff of an arithmetic Asian option in terms of a portfolio of European options. Moreover, it is shown that the obtained hedge is optimal in some sense. The strategy is based on stop-loss transforms and is applicable under general stock price models. We focus on some popular Lévy models. Numerical illustrations of the hedging performance are given for various Lévy models calibrated to market data of the S&P 500.Comonotonicity; Data; Hedging; Market; Model; Models; Optimal; Options; Performance; Portfolio; Strategy;
FROM RUIN TO BANKRUPTCY FOR COMPOUND POISSON SURPLUS PROCESSES
In classical risk theory, the infinite-time ruin probability of a surplus process Ct is calculated as the probability of the process becoming negative at some point in time. In this paper, we consider a relaxation of the ruin concept to the concept of bankruptcy, according to which one has a positive surplus-dependent probability to continue despite temporary negative surplus. We study the resulting bankruptcy probability for the compound Poisson risk model with exponential claim sizes for different bankruptcy rate functions, deriving analytical results, upper and lower bounds as well as an efficient simulation method. Numerical examples are given and the results are compared with the classical ruin probabilities. Finally, it is illustrated how the analysis can be extended to study the discounted penalty function under this relaxed ruin criterio
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