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Parisian ruin probability : the De Vylder type approximation

Zdeb Martyna, Teuerle Marek A.

Mathematica Applicanda

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2020

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Vol. 48, no. 2

157--171

EN

The Parisian ruin occurs as the capital of the insurance company is negative longer than a predefined period of time. In this article, we propose a simple and fast technique for calculating the Parisian ruin probability for the Cramér-Lundberg model with arbitrary claims that have the first three moments finite. The introduced method is based on the idea of the De Vylder approximation. We apply the method for various claim distributions and verify its accuracy. Lastly, the method is applied to a model that is fitted to the empirical data.

PL

Moment wystąpienia ruiny typu paryskiego definiowany jest jako moment, w którym kapitały firmy ubezpieczeniowej w czasie były w sposób ciągły ujemne przez wcześniej zdefiniowany okres czasu. W tej pracy zaproponowano szybką i relatywnie prostą technikę wyznaczania przybliżonego prawdopodbieństwa ruiny paryskiej dla procesu Craméra-Lundberga ze szkodami z dowolnego rozkładu, dla którego pierwsze trzy momenty są skończone. Przedstawiona metoda wykorzystuje metodę De Vyldera. W pracy sprawdzono też dokładność metody dla procesu Craméra-Lundberga o wybranych rozkładach szkód. Dodatkowo, wyznaczono prawdopodbieństwo ruiny paryskiej dla modelu opisującego dane empiryczne towarzystwa ubezpieczeniowego.

2

Ruin probability in a risk model with variable premium intensity and risky investments

Mishura Y., Perestyuk M., Ragulina O.

Opuscula Mathematica

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2015

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Vol. 35, no. 3

333--352

EN

We consider a generalization of the classical risk model when the premium intensity depends on the current surplus of an insurance company. All surplus is invested in the risky asset, the price of which follows a geometric Brownian motion. We get an exponential bound for the infinite-horizon ruin probability. To this end, we allow the surplus process to explode and investigate the question concerning the probability of explosion of the surplus process between claim arrivals.

3

On optimal stopping of risk processes with regime switching

Ferenstein E., Pasternak Winiarski A.

Demonstratio Mathematica

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2012

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Vol. 45, nr 2

435-454

EN

In the paper we solve a problem of optimal stopping of a risk process in two alternative settings. We assume that the main characteristics of the risk process change according to unobservable random variable. In the first model we assume that the post-disorder distributions are not known a'priori and are randomly chosen from a finite set of admissible distributions. The second model concentrates on a situation when more than one disorder is possible. For both models optimal stopping rules with respect to given utility function are constructed using dynamic programming methodology.

4

Dependent discrete risk processes –calculation of the probability of ruin

Heilpern S.

Operations Research and Decisions

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2010

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Vol. 20, No. 2

59-76

EN

This paper is devoted to discrete processes of dependent risks. The random variables describing the time between claims can be dependent in such processes, unlike under the classical approach. The ruin problem is investigated and the probably of ruin is computed. The relation between the degree of dependence and the probability of ruin is studied. Three cases are presented. Different methods of characterizing the dependency structure are examined. First, strictly dependent times between claims are investigated. Next, the dependency structure is described using an Archimedean copula or using Markov chains. In the last case, three situations in which the probability of ruin can be exactly computed are presented. Numerical examples in which the claims have a geometric distribution are investigated. A regular relation between the probability of ruin and the degree of dependence is only observed in the Markov chain case.

5

Optimal stopping of a risk process, an overview

Muciek B.

Prace Naukowe Instytutu Matematyki Politechniki Wrocławskiej. Konferencje

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2001

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Vol. 24, nr 3

91-104

EN

A classical problem in risk theory is considered. An insurance company receives premiums and pays out claims which are involved in a risk process. Some approaches to the problem of optimal stopping of the risk process are presented and some generalizations are brought up. Two similar models are solved in two different ways. One of them is by smooth semimartingale decomposition of the net gain, while the other is by solving dynamic programming equations. Both of them are generalized.

6

On nonzero-sum stopping game related to discrete risk processes

Bobecka K., Ferenstein E.

Control and Cybernetics

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2001

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Vol. 30, no 3

339-354

EN

We consider two-person nonzero-sum stopping game. The players (insurers) observe discrete time risk processes until one of them decides to stop his process. Strategies of the players are stopping times. The aim of each player is to maximize his expected gain. We find Nash equilibrium point for this game under certain assumptions on reward sequences.