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January  2017, 13(1): 207-222. doi: 10.3934/jimo.2016012

The finite-time ruin probability for an inhomogeneous renewal risk model

Emilija Bernackaitė and  Jonas Šiaulys ,

Faculty of Mathematics and and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania

* Corresponding author: jonas.siaulys@mif.vu.lt

Received  March 2015 Revised  December 2015 Published  March 2016

Fund Project: The second author is supported by a grant No. MIP-13079 from the Research Council of Lithuania.

In the paper, we give an asymptotic formula for the finite-time ruin probability in a generalized renewal risk model. We consider the renewal risk model with independent strongly subexponential claim sizes and independent not necessarily identically distributed inter occurrence times having finite variances. We find out that the asymptotic formula for the finite-time ruin probability is insensitive to the homogeneity of inter-occurrence times.

Keywords: Inhomogeneous renewal risk model, finite-time ruin probability, strongly subexponential distribution, asymptotic formula.
Mathematics Subject Classification: Primary: 91B30; Secondary: 60K05, 60G20.
Citation: Emilija Bernackaitė, Jonas Šiaulys. The finite-time ruin probability for an inhomogeneous renewal risk model. Journal of Industrial and Management Optimization, 2017, 13 (1) : 207-222. doi: 10.3934/jimo.2016012
References:
[1]

E. Sparre Andersen, On the collective theory of risk in case of contagion between claims, Transactions of the XV-th International Congress of Actuaries, 2 (1957).

[2]

I. M. Andrulytė, E. Bernackaitė, D. Kievinaitė and J. Šiaulys, A Lundberg-type inequality for an inhomogeneous renewal risk model, Modern Stochastics: Theory and Applications, 2 (2015), 173-184. 

[3]

E. Bernackaitė and J. Šiaulys, The exponential moment tail of inhomogeneous renewal process, Statistics and Probability Letters, 97 (2015), 9-15.  doi: 10.1016/j.spl.2014.10.018.

[4]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511721434.

[5]

V. P. Chistyakov (Čistjakov), A theorem on sums of independent positive random variables and its applications to branching processes, Theory of Probability and Its Applications (Teoriya Veroyatnostei i ee Primeneniya), 9 (1964), 640-648. 

[6]

P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer, New York, 1997. doi: 10.1007/978-3-642-33483-2.

[7]

P. Embrechts and N. Veraverbeke, Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insurance: Mathematics and Economics, 1 (1984), 55-72.  doi: 10.1016/0167-6687(82)90021-X.

[8]

S. Foss, D. Korshunov and S. Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions, Springer, 2011. doi: 10.1007/978-1-4419-9473-8.

[9]

R. Kaas and Q. Tang, Note on the tail behavior of random walk maxima with heavy tails and negative drift, North American Actuarial Journal, 7 (2003), 57-61.  doi: 10.1080/10920277.2003.10596103.

[10]

J. Kočetova, R. Leipus and J. Šiaulys, A property of the renewal counting process with application to the finite-time ruin probability, Lithuanian Mathematical Journal, 49 (2009), 55-61.  doi: 10.1007/s10986-009-9032-1.

[11]

D. Korshunov, Large-deviation probabilities for maxima of sums of independent random variables with negative mean and subexponential distribution, Theory of Probability and its Applications, 46 (2002), 355-366.  doi: 10.1137/S0040585X97979019.

[12]

R. Leipus and J. Šiaulys, Asymptotic behaviour of the finite-time ruin probability in renewal risk models, Applied Stochastic Models in Bussines and Industry, 25 (2009), 309-321.  doi: 10.1002/asmb.747.

[13]

V. V. Petrov, Limit Theorems of Probability Theory, Clarendon Press, Oxford, 1995.

[14]

E. J. G. Pitman, Subexponential distribution functions, Journal of Australian Mathematical Society (Series A), 29 (1980), 337-347.  doi: 10.1017/S1446788700021340.

[15]

A. N. Shiryaev, Probability, Springer, 1996. doi: 10.1007/978-1-4757-2539-1.

[16]

W. L. Smith, On the elementary renewal theorem for non -identicaly distributed variables, Pacific Journal of Mathematics, 14 (1964), 673-699.  doi: 10.2140/pjm.1964.14.673.

[17]

Q. Tang, Asymptotics for the finite time ruin probability in the renewal model with consistent variation, Stochastic Models, 20 (2004), 281-297.  doi: 10.1081/STM-200025739.

[18]

N. Veraverbeke, Asymptotic behavior of Wiener-Hopf factors of a random walk, Stochastic Processes and their Applications, 5 (1977), 27-37.  doi: 10.1016/0304-4149(77)90047-3.

[19]

Y. Wang, Z. Cui, K. Wang and X. Ma, Uniform asymptotics of the finite-time ruin probability for all times, Journal of Mathematical Analysis and Applications, 390 (2012), 208-223.  doi: 10.1016/j.jmaa.2012.01.025.

show all references

References:
[1]

E. Sparre Andersen, On the collective theory of risk in case of contagion between claims, Transactions of the XV-th International Congress of Actuaries, 2 (1957).

[2]

I. M. Andrulytė, E. Bernackaitė, D. Kievinaitė and J. Šiaulys, A Lundberg-type inequality for an inhomogeneous renewal risk model, Modern Stochastics: Theory and Applications, 2 (2015), 173-184. 

[3]

E. Bernackaitė and J. Šiaulys, The exponential moment tail of inhomogeneous renewal process, Statistics and Probability Letters, 97 (2015), 9-15.  doi: 10.1016/j.spl.2014.10.018.

[4]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511721434.

[5]

V. P. Chistyakov (Čistjakov), A theorem on sums of independent positive random variables and its applications to branching processes, Theory of Probability and Its Applications (Teoriya Veroyatnostei i ee Primeneniya), 9 (1964), 640-648. 

[6]

P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer, New York, 1997. doi: 10.1007/978-3-642-33483-2.

[7]

P. Embrechts and N. Veraverbeke, Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insurance: Mathematics and Economics, 1 (1984), 55-72.  doi: 10.1016/0167-6687(82)90021-X.

[8]

S. Foss, D. Korshunov and S. Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions, Springer, 2011. doi: 10.1007/978-1-4419-9473-8.

[9]

R. Kaas and Q. Tang, Note on the tail behavior of random walk maxima with heavy tails and negative drift, North American Actuarial Journal, 7 (2003), 57-61.  doi: 10.1080/10920277.2003.10596103.

[10]

J. Kočetova, R. Leipus and J. Šiaulys, A property of the renewal counting process with application to the finite-time ruin probability, Lithuanian Mathematical Journal, 49 (2009), 55-61.  doi: 10.1007/s10986-009-9032-1.

[11]

D. Korshunov, Large-deviation probabilities for maxima of sums of independent random variables with negative mean and subexponential distribution, Theory of Probability and its Applications, 46 (2002), 355-366.  doi: 10.1137/S0040585X97979019.

[12]

R. Leipus and J. Šiaulys, Asymptotic behaviour of the finite-time ruin probability in renewal risk models, Applied Stochastic Models in Bussines and Industry, 25 (2009), 309-321.  doi: 10.1002/asmb.747.

[13]

V. V. Petrov, Limit Theorems of Probability Theory, Clarendon Press, Oxford, 1995.

[14]

E. J. G. Pitman, Subexponential distribution functions, Journal of Australian Mathematical Society (Series A), 29 (1980), 337-347.  doi: 10.1017/S1446788700021340.

[15]

A. N. Shiryaev, Probability, Springer, 1996. doi: 10.1007/978-1-4757-2539-1.

[16]

W. L. Smith, On the elementary renewal theorem for non -identicaly distributed variables, Pacific Journal of Mathematics, 14 (1964), 673-699.  doi: 10.2140/pjm.1964.14.673.

[17]

Q. Tang, Asymptotics for the finite time ruin probability in the renewal model with consistent variation, Stochastic Models, 20 (2004), 281-297.  doi: 10.1081/STM-200025739.

[18]

N. Veraverbeke, Asymptotic behavior of Wiener-Hopf factors of a random walk, Stochastic Processes and their Applications, 5 (1977), 27-37.  doi: 10.1016/0304-4149(77)90047-3.

[19]

Y. Wang, Z. Cui, K. Wang and X. Ma, Uniform asymptotics of the finite-time ruin probability for all times, Journal of Mathematical Analysis and Applications, 390 (2012), 208-223.  doi: 10.1016/j.jmaa.2012.01.025.

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