Modern Stochastics: Theory and Applications logo


  • Help
Login Register

  1. Home
  2. Issues
  3. Volume 2, Issue 2 (2015)
  4. A Lundberg-type inequality for an inhomo ...

Modern Stochastics: Theory and Applications

Submit your article Information Become a Peer-reviewer
  • Article info
  • Full article
  • Related articles
  • Cited by
  • More
    Article info Full article Related articles Cited by

A Lundberg-type inequality for an inhomogeneous renewal risk model
Volume 2, Issue 2 (2015), pp. 173–184
Ieva Marija Andrulytė   Emilija Bernackaitė   Dominyka Kievinaitė   Jonas Šiaulys  

Authors

 
Placeholder
https://doi.org/10.15559/15-VMSTA30
Pub. online: 31 July 2015      Type: Research Article      Open accessOpen Access

Received
19 June 2015
Revised
25 July 2015
Accepted
26 July 2015
Published
31 July 2015

Abstract

We obtain a Lundberg-type inequality in the case of an inhomogeneous renewal risk model. We consider the model with independent, but not necessarily identically distributed, claim sizes and the interoccurrence times. In order to prove the main theorem, we first formulate and prove an auxiliary lemma on large values of a sum of random variables asymptotically drifted in the negative direction.

References

[1] 
Albrecher, H., Teugels, J.L.: Exponential behavior in the presence of dependence in risk theory. J. Appl. Probab. 43(1), 257–273 (2006). MR2225065. doi:10.1239/jap/1143936258
[2] 
Andersen Sparre, E.: On the collective theory of risk in the case of contagion between the claims. In: Transactions XVth International Congress of Actuaries, vol. 2(6), pp. 219–229 (1957)
[3] 
Asmussen, S., Albrecher, H.: Ruin Probabilities. World Scientific Publishing (2010). MR2766220. doi:10.1142/9789814282536
[4] 
Bernackaitė, E., Šiaulys, J.: The exponential moment tail of inhomogeneous renewal process. Stat. Probab. Lett. 97, 9–15 (2015). MR3299745. doi:10.1016/j.spl.2014.10.018
[5] 
Bieliauskienė, E., Šiaulys, J.: Infinite time ruin probability in inhomogeneous claim case. Liet. Mat. Rink., LMD Darbai 51, 352–356 (2010) (ISSN 0132-2818, available at page www.mii.lt/LMR/)
[6] 
Blaževičius, K., Bieliauskienė, E., Šiaulys, J.: Finite-time ruin probability in the inhomogeneous claim case. Lith. Math. J. 50(3), 260–270 (2010). MR2719562. doi:10.1007/s10986-010-9084-2
[7] 
Chen, Y., Ng, K.W.: The ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims. Insur. Math. Econ. 40(3), 415–423 (2007). MR2310980. doi:10.1016/j.insmatheco.2006.06.004
[8] 
Cramér, H.: Historical review of Filip Lundberg’s works on risk theory. Scand. Actuar. J. 1969, 6–12 (1969). MR0347028. doi:10.1080/03461238.1969.10404602
[9] 
Embrechts, P., Klüppelberg, C., Mikosch, T.: Modeling Extremal Events. Springer (1997). MR1458613. doi:10.1007/978-3-642-33483-2
[10] 
Embrechts, P., Veraverbeke, N.: Estimates for probability of ruin with special emphasis on the possibility of large claims. Insur. Math. Econ. 1(1), 55–72 (1982). MR0652832. doi:10.1016/0167-6687(82)90021-X
[11] 
Gerber, H.U.: Martingales in risk theory. Mitt., Schweiz. Ver. Versicher.math. 73, 205–216 (1973)
[12] 
Lefèvre, C., Picard, P.: A nonhomogeneous risk model for insurance. Comput. Math. Appl. 51, 325–334 (2006). MR2203083. doi:10.1016/j.camwa.2005.11.005
[13] 
Li, J., Tang, Q., Wu, R.: Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model. Adv. Appl. Probab. 42(4), 1126–1146 (2010). MR2796679. doi:10.1239/aap/1293113154
[14] 
Lundberg, F.: Approximerad framställning av sannolikhetsfunktionen. Återförsäkring av kollektivrisker. Acad. Afhaddling. Almqvist. och Wiksell, Uppsala (1903)
[15] 
Lundberg, F.: Some supplementary researches on the collective risk theory. Skand. Aktuarietidskr. 15, 137–158 (1932)
[16] 
Mikosch, T.: Non-life Insurance Mathematics. Springer (2009). MR2503328. doi:10.1007/978-3-540-88233-6
[17] 
Raducan, A.M., Vernic, R., Zbaganu, G.: Recursive calculation of ruin probabilities at or before claim instants for non-identically distributed claims. ASTIN Bull. 45, 421–443 (2015). doi:10.1017/asb.2014.30
[18] 
Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.: Stochastic Process for Insurance and Finance. John Wiley & Sons Ltd (1998). MR1680267. doi:10.1002/9780470317044
[19] 
Sgibnev, M.S.: Submultiplicative moments of the suppremum of a random walk with negative drift. Stat. Probab. Lett. 32, 377–383 (1997). MR1602211. doi:10.1016/S0167-7152(96)00097-1
[20] 
Smith, W.L.: On the elementary renewal theorem for non-identically distributed variables. Pac. J. Math. 14(2), 673–699 (1964). MR0165598. doi:10.2140/pjm.1964.14.673
[21] 
Teugels, J., Sund, B. (eds.): Enciklopedia of Actuarial Science. Wiley (2004)
[22] 
Thorin, O.: Some comments on the Sparre Andersen model in the risk theory. ASTIN Bull. 8(1), 104–125 (1974). MR0351020
[23] 
Wang, K., Wang, Y., Gao, Q.: Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate. Methodol. Comput. Appl. Probab. 15(1), 109–124 (2013). MR3030214. doi:10.1007/s11009-011-9226-y

Full article Related articles Cited by PDF XML
Full article Related articles Cited by PDF XML

Copyright
© 2015 The Author(s). Published by VTeX
by logo by logo
Open access article under the CC BY license.

Keywords
Inhomogeneous model renewal model Lundberg-type inequality exponential bound ruin probability

MSC2010
91B30 60G50

Metrics
since March 2018
1857

Article info
views

496

Full article
views

420

PDF
downloads

171

XML
downloads

Export citation

Copy and paste formatted citation
Placeholder

Download citation in file


Share


RSS

MSTA

MSTA

  • Online ISSN: 2351-6054
  • Print ISSN: 2351-6046
  • Copyright © 2018 VTeX

About

  • About journal
  • Indexed in
  • Editors-in-Chief

For contributors

  • Submit
  • OA Policy
  • Become a Peer-reviewer

Contact us

  • ejournals-vmsta@vtex.lt
  • Mokslininkų 2A
  • LT-08412 Vilnius
  • Lithuania
Powered by PubliMill  •  Privacy policy