ON APPLICATION OF RANDOM WALK WITH DELAY AND PARETO DISTRIBUTED INTERFERENCE OF CHANCE TO AN INSURANCE MODEL

Year 2016, Volume: 29 Issue: 3, 615 - 626, 30.09.2016

Zafer Küçük Tülay Kesemen Tahir Khaniyev Fuat Yetim Aslı Bektaş Kamışlık

Abstract

In this study, a semi-Markovian random walk (X(t)) with Pareto distributed interference of chance and delay is considered. Some exact formulas for the first four stationary moments of the process X(t) are obtained, when the random variables $\zetta_n,n=1,2,...$ which express the discrete interference of chance have Pareto distribution with parameters $(\alpha, \lamda)$. The random variables $\zetta_n,n=1,2,...$ are interpreted as loans which insurance company gets from a bank. With the use of these exact formulas, the third-order asymptotic expansions for the first four stationary moments of the process X(t) are derived, when $E(zetta_n)$ is sufficiently large. Finally, by using Monte-Carlo simulation method, the accuracy of obtained approximation formulas is tested.

Keywords

Insurance model, random walk with delay, Pareto distribution, asymptotic expansion, Monte-Carlo simulation method

References

• Aliyev, R.T., Khaniyev, T.A. and Kesemen, T., “Asymptotic expansions for the moments of a semi-Markovian random walk with gamma distributed interference of chance”, Communications in Statistics- Theory and Methods, 39(1): 130-143, (2010).
• Aliyev, R., Kucuk, Z. and Khaniyev, T., “Three-term asymptotic expansions for the moments of the random walk with triangular distributed interference of chance”, Applied Mathematical Modelling, 34(11):3599-3607, (2010).
• Anisimov, V.V. and Artalejo, J.R., “Analysis of Markov multiserver retrial queues with negative arrivals”, Queuing Systems: Theory and Applic., 39(2/3):157–182, (2001).
• Borovkov, A.A., Stochastic Process in Queuing Theory, Springer, New York, (1976).
• Feller, W., Introduction to Probability Theory and Its Appl. II, New York, (1971)
• Gihman, I.I. and Skorohod, A.V., Theory of Stochastic Processes II, Berlin, (1975).
• Khaniyev, T.A. and Mammadova, Z. “On the stationary characteristics of the extended model of type (s,S) with Gaussian distribution of summands”, Journal of Statistical Computation and Simulation, 76(10): 861–874 , (2006).
• Khaniyev, T.A., Kesemen, T., Aliyev, R.T. and Kokangul, A., “Asymptotic expansions for the moments of a semi-Markovian random walk with exponential distributed interference of chance”, Statistics & Probability Letters, 78(6):785–793, (2008).
• Lotov, V.I., “On some boundary crossing problems for Gaussian random walks”, The Annals of Probability, 24(4): 2154–2171, (1996).
• Rogozin, B.A., “On the distribution of the first jump”, Theory Probability and Its Applications, 9(3):498–545, (1964).