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Simultaneous ruin probability for two-dimensional brownian risk model

Part of: Stochastic processes

Published online by Cambridge University Press:  16 July 2020

Krzysztof Dȩbicki ,
Enkelejd Hashorva  and
Zbigniew Michna
Krzysztof Dȩbicki*
Affiliation:
University of Wrocław
Enkelejd Hashorva*
Affiliation:
University of Lausanne
Zbigniew Michna*
Affiliation:
Wrocław University of Economics and Business
*
*Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email: Krzysztof.Debicki@math.uni.wroc.pl
**Postal address: Department of Actuarial Science, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland. Email: Enkelejd.Hashorva@unil.ch
***Postal address: Department of Logistics, Wrocław University of Economics and Business, Komandorska 118/120, 53-345 Wrocław, Poland. Email: zbigniew.michna@ue.wroc.pl
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Abstract

The ruin probability in the classical Brownian risk model can be explicitly calculated for both finite and infinite time horizon. This is not the case for the simultaneous ruin probability in the two-dimensional Brownian risk model. Relying on asymptotic theory, we derive in this contribution approximations for both simultaneous ruin probability and simultaneous ruin time for the two-dimensional Brownian risk model when the initial capital increases to infinity.

Keywords

two-dimensional Brownian risk modelBrownian motionsimultaneous ruin probabilitysimultaneous ruin timeruin time approximation

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 2 , June 2020 , pp. 597 - 612
Copyright
© Applied Probability Trust 2020

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References

Asmussen, S., Hashorva, E., Laub, P. J. andTaimre, T. (2017). Tail asymptotics of light-tailed Weibull-like sums. Prob. Math. Statist. 37, 235–256.10.19195/0208-4147.37.2.3CrossRefGoogle Scholar
Avram, F., Palmowski, Z. andPistorius, M. R. (2008). A two-dimensional ruin problem on the positive quadrant. Insurance Math. Econom. 42, 227–234.10.1016/j.insmatheco.2007.02.004CrossRefGoogle Scholar
Avram, F., Palmowski, Z. andPistorius, M. R. (2008). Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results. Ann. Appl. Prob. 18, 2421–2449.10.1214/08-AAP529CrossRefGoogle Scholar
Bai, L., Dȩbicki, K. andLiu, P. (2018). Extremes of vector-valued Gaussian processes with trend. J. Math. Anal. Appl. 465, 47–74.10.1016/j.jmaa.2018.04.069CrossRefGoogle Scholar
Borovkov, K. andPalmowski, Z. (2019). The exact asymptotics for hitting probability of a remote orthant by a multivariate Lévy process: The Cramér case. In 2017 MATRIX Annals (MATRIX Book Series 2), eds D. Wood, J. de Gier, C. Praeger, and T. Tao. Springer, Cham.10.1007/978-3-030-04161-8_20CrossRefGoogle Scholar
Dȩbicki, K. andHashorva, E. (2017). On extremal index of max-stable stationary processes. Prob. Math. Statist. 37, 299–317.Google Scholar
Dȩbicki, K., Hashorva, E., Ji, L. andRolski, T. (2018). Extremal behavior of hitting a cone by correlated Brownian motion with drift. Stoch. Process. Appl. 128, 4171–4206.10.1016/j.spa.2018.02.002CrossRefGoogle Scholar
Dȩbicki, K. andMandjes, M. (2015). Queues and Lévy Fluctuation Theory. Springer, New York.10.1007/978-3-319-20693-6CrossRefGoogle Scholar
Delsing, G. A., Mandjes, M., Spreij, P. J. andWinands, E. M. (2018). Asymptotics and approximations of ruin probabilities for multivariate risk processes in a Markovian environment. Preprint, .Google Scholar
Dombry, C. andRabehasaina, L. (2017). High order expansions for renewal functions and applications to ruin theory. Ann. Appl. Prob. 27, 2342–2382.Google Scholar
Foss, S., Korshunov, D., Palmowski, Z. andRolski, T. (2017). Two-dimensional ruin probability for subexponential claim size. Prob. Math. Statist. 37, 319–335.Google Scholar
Furrer, H, Michna, Z. andWeron, A. (1997). Stable Lévy motion approximation in collective risk theory. Insurance Math. Econom. 20, 97–114.10.1016/S0167-6687(97)00008-5CrossRefGoogle Scholar
Hashorva, E. (2019). Approximation of some multivariate risk measures for Gaussian risks. J. Multivar. Anal. 169, 330–340.10.1016/j.jmva.2018.10.006CrossRefGoogle Scholar
Hu, Z. andJiang, B. (2013). On joint ruin probabilities of a two-dimensional risk model with constant interest rate. J. Appl. Prob. 50, 309–322.10.1239/jap/1371648943CrossRefGoogle Scholar
Iglehart, D. (1969). Diffusion approximations in collective risk theory. J. Appl. Prob. 6, 285–292.10.2307/3211999CrossRefGoogle Scholar
Ji, L. andRobert, S. (2018). Ruin problem of a two-dimensional fractional Brownian motion risk process. Stoch. Models 34, 73–97.10.1080/15326349.2017.1389284CrossRefGoogle Scholar
Korshunov, D. andWang, L. (2019). Tail asymptotics for Shepp-statistics of Brownian motion in $R^d$ . Extremes 23, 35–54.10.1007/s10687-019-00357-zCrossRefGoogle Scholar
Lieshout, P. andMandjes, M. (2007). Tandem Brownian queues. Math. Meth. Operat. Res. 66, 275–298.10.1007/s00186-007-0149-xCrossRefGoogle Scholar
Lifshits, M. (2012). Lectures on Gaussian Processes (Springer Briefs in Mathematics). Springer, Heidelberg.Google Scholar
Michna, Z. (2017). Remarks on Pickands’ theorem. Prob. Math. Statist. 37, 373–393.10.19195/0208-4147.37.2.10CrossRefGoogle Scholar
Michna, Z. (2018). Ruin probabilities for two collaborating insurance companies. Preprint, . To appear in Prob. Math. Statist.Google Scholar
Pan, Y. andBorovkov, K. A. (2019). The exact asymptotics of the large deviation probabilities in the multivariate boundary crossing problem. Adv. Appl. Prob. 51, 835–864.10.1017/apr.2019.28CrossRefGoogle Scholar
Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields (Translations of Mathematical Monographs 148). American Mathematical Society, Providence, RI.Google Scholar
Piterbarg, V. I. (2015). Twenty Lectures About Gaussian Processes. Atlantic Financial Press, London.Google Scholar
Samorodnitsky, G. andSun, J. (2016). Multivariate subexponential distributions and their applications. Extremes 19, 171–196.10.1007/s10687-016-0242-8CrossRefGoogle Scholar