**The Model of
Ruin Probability of Insurance Company**

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*Sergiy
Illichevskyy,*

*Postgraduate student of the Taras Shevchenko National
University of Kyiv, Ukraine.*

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**Модель вероятности банкротства страховой компании**

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*Илличевский Сергей Алексеевич,*

*аспирант Киевского национального университета имени
Тараса Шевченко.*

* *

Nowadays it is impossible to imagine a market economy without risks. They are involved almost in every economic activity. There is a great need in measuring, predicting and minimizing risks. Insurance services are one of the industries, which permanently experience risks of bankruptcy. That is why calculating the ruin probabilities for insurance companies are one of the problems that need well-developed mathematical models [1, p. 27-32].

One of the first studies in this area was conducted in the beginning of the twentieth century. Since then, the mathematical methods of ruin probability calculation developed and accumulated a great variety of models and approaches. While the permanent growing of economic needs, insurance services increase steadily in the economies of all developed countries. Insurance services are one of the youngest industries any economy, which experience a stage of active development.

We consider an insurance company in the case when the premium rate is a bounded by some nonnegative random function and the capital of the insurance company is invested in a risky asset whose price follows a geometric Brownian.

There are different methods for approximating the distribution of aggregate claims and their corresponding stop-loss premium by means of a discrete compound Poisson distribution and its corresponding stop-loss premium. This discretization is an important step in the numerical evaluation of the distribution of aggregate claims, because recent results on recurrence relations for probabilities only apply to discrete distributions. The discretization technique is efficient in a certain sense, because a properly chosen discretization gives raise to numerical upper and lower bounds on the stop-loss premium, giving the possibility of calculating the numerically estimates for the error on the final numerical results.

См. насос для керосина 12в https://www.personalazs.ru/ .

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We consider an insurance company in the case when the premium rate is some bounded nonnegative random function and the capital of the insurance company is invested in a risky asset whose price follows a geometric Brownian motion with some mean return and volatility. We may find exact the asymptotic upper and lower bounds for the ruin probability as the initial endowment tends to infinity. We investigate the problem of consistency of risk measures with respect to usual stochastic order and convex order.

It is well-known that the analysis of activity of an insurance company in conditions of uncertainty is of great importance [2, p. 14-17]. Starting from the classical papers of Cramer and Lundberg which first considered the ruin problem in stochastic environment, this subject has attracted much attention. Recall that, in the classical Cramer-Lundberg model satisfying the Cramer condition and, the positive safety loading assumption, the ruin probability as a function of the initial endowment decreases exponentially [3, p. 47-48]. The problem was subsequently extended to the case when the insurance risk process is a general Levy process.

More recently ruin problems have been studied in application to an insurance company which invests its capital in a risky asset see, e.g., Paulsen [4, p. 136-140], Kalshnikov and Norberg [5, p. 220-223], Frolova, Kabanov, Pergamenshchikov [6, p. 229-232] and many others.

It is clear that, risky investment can be dangerous: disasters may arrive in the period when the market value of assets is low and the company will not be able to cover losses by selling these assets because of price fluctuations. Regulators are rather attentive to this issue and impose stringent constraints on company portfolios.

We deal with the ruin problem for an insurance company investing its capital in a risky asset specified by a geometric Brownian motion

In practice this means that the company should obtain a premium with the same rate continuously. We think that this condition is too restrictive and it significantly bounds the applicability of the above mentioned results in practical insurance settings.

Our goal is to consider the ruin problem for an insurance company for which the premium rate is specified by a bounded non-negative random function. For the given problem, under the condition of small volatility, we derive exact upper and lower bounds for the ruin probability and in the case of exponential premium rate, so we find the exact asymptotics for the ruin probability. Particularly, we can show that for the zero premium rates, the asymptotic result is the same as in the case of negative premium rates.

**References**

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1. Страхування: Підручник / За ред. В.Д. Базилевича. – К.: Знання-Прес, 2008. – 1019 с.

2. Анісімов В.В., Черняк О.І. Математична статистика: Навч. посібник для студентів вузів.− К.:МП „Олеся”, 1995. −104 с.

3. Гихман И.И., Скороход А.В., Ядренко М.И. Теория вероятностей и математическая статистика. − Київ: Вища школа, 1998.

4. Kalashnikov, V., Norberg, R. (2002) Power tailed ruin probabilities in the presence of risky investments. Stochastic Process. Appl. 98 211-228.

5. Paulsen, J. (1998) Sharp conditions for certain ruin in a risk process with stochastic return on investments. Stoch. Proc. Appl., 75 135-148.

6. Frolova, A.G., Kabanov, Yu.M. and Pergamenshchikov, S.M. (2002) In the insurance business risky investments are dangerous. Finance and Stochas-tics., 6 227-235.

**Поступила
в редакцию 05.12.2011 г.**