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1
Content available remote Exponential bounds of ruin probabilities for non-homogeneous risk models
Zhou Qianqian, Sakhanenko Alexander, Guo Junyi
Probability and Mathematical Statistics
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2021
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Vol. 41, Fasc. 2
17--235
EN
Lundberg-type inequalities for ruin probabilities of non-homogeneous risk models are presented. By employing the martingale method, upper bounds of ruin probabilities are obtained for general risk models under weak assumptions. In addition, several risk models, including the newly defined united risk model and a quasi-periodic risk model with interest rate, are studied.
2
Content available remote Ruin Probabilities for Two Collaborating Insurance Companies
Michna Zbigniew
Probability and Mathematical Statistics
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2020
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Vol. 40, Fasc. 2
360--386
EN
We find a formula for the supremum distribution of spectrally positive or negative Lévy processes with a broken linear drift. This gives formulas for ruin probabilities if two insurance companies (or two branches of the same company) divide between them both claims and premia in some specified proportions or if the premium rate for a given insurance portfolio is changed at a certain time. As an example we consider a gamma Lévy process, an -stable Lévy process and Brownian motion. Moreover we obtain identities for the Laplace transform of the distribution of the supremum of Lévy processes with a randomly broken drift (random time of the premium rate change) and on random intervals (random time when the insurance portfolio is closed).
3
Content available remote De Vylder type approximation of the ruin probability for the insurer-reinsurer model
Burnecki Krzysztof, Teuerle Marek A., Wilkowska Aleksandra
Mathematica Applicanda
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2019
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Vol. 47, no. 1
5--24
EN
In this article we introduce a De Vylder type of approximation of the ruin probability for a two-dimensional risk process, where claims and premiums are shared with a predetermined proportion. Such a process is usually associated with the insurer-reinsurer model. Applying De Vylder's idea to the risk process we obtain an approximation of the ruin probability for an arbitrary claim amount distribution Orly assuming that the third moment exists. We check performance of the approximation by means of the Monte Carlo simulations studying several typical claim Mount distributions. All results show that the proposed approximation yields very small relative errors. Finally, we illustrate the approximation by considering real-world loss data obtained from a Polish insurance company.
PL
W niniejszej pracy rozważano problem aproksymacji prawdopodobieństwa ruiny dla dwuwymiarowego procesu ryzyka, dla którego wszystkie szkody jak i zebrana składka dzielone są pomiędzy dwa składowe procesy ryzyka wg wcześniej zdefiniowanej i stałej w czasie proporcji. Taki proces ryzyka może opisywać układ ubezpieczyciela i reasekuratora, dla których wszystkie polisy w portfelu objęte są kontraktem reasekuracji proporcjonalnej. Stosując technikę zaproponowaną przez De Vyldera uzyskano aproksymację prawdopodobieństwa ruiny w przypadku, gdy szkody w rozważanym dwuwymiarowym procesie ryzyka są z dowolnego rozkładu o skończonych pierwszych trzech momentach. Jakość uzyskanych przybliżeń prawdopodobieństwa ruiny została zweryfikowana za pomocą symulacji Monte Carlo dla kilku typowych rozkładów prawdopodobieństwa używanych do modelowania szkód ubezpieczeniowych. Wszystkie wyniki wskazują, że zaproponowana aproksymacja prowadzi do małych błędów względnych. Na koniec, opracowana technika została użyta dla rzeczywistych danych uzyskanych od jednego z polskich towarzystw ubezpieczeniowych.
4
Content available remote Two-dimensional ruin probability for subexponential claim size
Foss S., Korshunov D., Palmowski Z., Rolski T.
Probability and Mathematical Statistics
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2017
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Vol. 37, Fasc. 2
319--335
EN
We analyse the asymptotics of ruin probabilities of two insurance companies (or two branches of the same company) that divide between them both claims and premiums in some specified proportions when the initial reserves of both companies tend to infinity, and generic claim size is subexponential.
5
Content available Ruin probability of a discrete-time risk process with proportional reinsurance and investment for exponential and Pareto distributions
Jasiulewicz H., Kordecki W.
Operations Research and Decisions
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2015
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Vol. 25, No. 3
17--38
EN
The paper focuses on a quantitative analysis of the probability of ruin in a finite time for a discrete risk process with proportional reinsurance and investment of the financial surplus. It is assumed that the total loss on a unit interval has either a light-tailed distribution – exponential distribution or a heavytailed distribution – Pareto distribution. The ruin probabilities for the finite-horizons 5 and 10 were determined from recurrence equations. Moreover, the upper bound of the ruin probability is given for the exponential distribution based on the Lundberg adjustment coefficient. This adjustment coefficient does not exist for the Pareto distribution, hence an asymptotic approximation is given for the ruin probability when the initial capital tends to infinity. The numerical results obtained are illustrated by tables and figures.
6
Content available remote Probability of failure with discrete claim distribution
Iwanik J.
Probability and Mathematical Statistics
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2005
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Vol. 25, Fasc. 2
307--316
EN
A concept of the probability of failure is introduced. Some popular methods of exact computation of ruin probability are adopted to compute failure probability. Based on the formula presented in [5] a generalization of a ruin probability algorithm is proposed that can also be used for failure probability. The algorithm’s computational complexity is studied and it is proved to be more effective for failure probability than for ruin probability. Finally, some numerical examples for failure probability computations are given.
7
Content available remote Tail probabilities for a risk process with subexponential jumps in a regenerative and diffusion environment
Palmowski Z.
Probability and Mathematical Statistics
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2002
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Vol. 22, Fasc. 2
381-405
EN
In this paper we find a nonexponential Lundberg approximation of the ruin probability in a Cox model, in which a governing process has a regenerative structure and claims are light-tailed or have an intermediate regularly varying distribution. Examples include an intensity process being reflected Brownian motion, square functions of the Ornstein-Uhlenbeck process and splitting reflected Brownian bridges. In particular, we consider a non-Markovian intensity proces.
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