Exact expression of ultimate time survival probability in homogeneous discrete-time risk model

Andrius Grigutis; Andrius Grigutis

doi:10.3934/math.2023260

AIMS Mathematics

2023, Volume 8, Issue 3: 5181-5199. doi: 10.3934/math.2023260

Previous Article Next Article

Research article

Exact expression of ultimate time survival probability in homogeneous discrete-time risk model

Andrius Grigutis ^,

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

Received: 04 September 2022 Revised: 15 November 2022 Accepted: 29 November 2022 Published: 13 December 2022
MSC : 60G50, 60J80, 91G05

In this work, we set up the generating function of the ultimate time survival probability $ \varphi(u+1) $, where

$ \varphi(u) = \mathbb{P}\left(\sup\limits_{n\geqslant 1}\sum\limits_{i = 1}^{n}\left(X_i- \kappa\right)<u\right), $

$ u\in\mathbb{N}_0, \, \kappa\in\mathbb{N} $ and the random walk $ \left\{\sum_{i = 1}^{n}X_i, \, n\in\mathbb{N}\right\} $ consists of independent and identically distributed random variables $ X_i $, which are non-negative and integer-valued. We also give expressions of $ \varphi(u) $ via the roots of certain polynomials. The probability $ \varphi(u) $ means that the stochastic process

$ u+ \kappa n-\sum\limits_{i = 1}^{n}X_i $

is positive for all $ n\in\mathbb{N} $, where a certain growth is illustrated by the deterministic part $ u+ \kappa n $ and decrease is given by the subtracted random part $ \sum_{i = 1}^{n}X_i $. Based on the proven theoretical statements, we give several examples of $ \varphi(u) $ and its generating function expressions, when random variables $ X_i $ admit Bernoulli, geometric and some other distributions.
- homogeneous discrete-time risk model,
- random walk,
- survival probability,
- initial values,
- generating function,
- Vandermonde matrix,
- ruin theory
Citation: Andrius Grigutis. Exact expression of ultimate time survival probability in homogeneous discrete-time risk model[J]. AIMS Mathematics, 2023, 8(3): 5181-5199. doi: 10.3934/math.2023260

Related Papers:

Abstract

In this work, we set up the generating function of the ultimate time survival probability $ \varphi(u+1) $, where

$ \varphi(u) = \mathbb{P}\left(\sup\limits_{n\geqslant 1}\sum\limits_{i = 1}^{n}\left(X_i- \kappa\right)<u\right), $

$ u\in\mathbb{N}_0, \, \kappa\in\mathbb{N} $ and the random walk $ \left\{\sum_{i = 1}^{n}X_i, \, n\in\mathbb{N}\right\} $ consists of independent and identically distributed random variables $ X_i $, which are non-negative and integer-valued. We also give expressions of $ \varphi(u) $ via the roots of certain polynomials. The probability $ \varphi(u) $ means that the stochastic process

$ u+ \kappa n-\sum\limits_{i = 1}^{n}X_i $

is positive for all $ n\in\mathbb{N} $, where a certain growth is illustrated by the deterministic part $ u+ \kappa n $ and decrease is given by the subtracted random part $ \sum_{i = 1}^{n}X_i $. Based on the proven theoretical statements, we give several examples of $ \varphi(u) $ and its generating function expressions, when random variables $ X_i $ admit Bernoulli, geometric and some other distributions.

References

[1]	S. M. Li, Y. Lu, J. Garrido, A review of discrete-time risk models, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat., 103 (2009), 321–337. https://doi.org/10.1007/BF03191910 doi: 10.1007/BF03191910
[2]	A. Grigutis, J. Šiaulys, Recurrent sequences play for survival probability of discrete time risk model, Symmetry, 12 (2020), 2111. https://doi.org/10.3390/sym12122111 doi: 10.3390/sym12122111
[3]	A. Grigutis, J. Jankauskas, On $2\times2$ determinants originating from survival probabilities in homogeneous discrete time risk model, Results Math., 77 (2022), 204. https://doi.org/10.1007/s00025-022-01736-y doi: 10.1007/s00025-022-01736-y
[4]	W. Feller, An introduction to probability theory and its applications. Vol. Ⅱ., 2 Ed., New York: John Wiley & Sons, 1971.
[5]	E. S. Andersen, On the collective theory of risk in case of contagion between the claims, Trans. XVth Int. Congr. Actuaries, 2 (1957), 219–229.
[6]	F. Spitzer, Principles of random walk, 2 Ed., New York: Springer, 2001.
[7]	H. U. Gerber, Mathematical fun with the compound binomial process, ASTIN Bull., 18 (1988), 161–168. https://doi.org/10.2143/AST.18.2.2014949 doi: 10.2143/AST.18.2.2014949
[8]	H. U. Gerber, Mathematical fun with ruin theory, Insur. Math. Econ., 7 (1988), 15–23. https://doi.org/10.1016/0167-6687(88)90091-1 doi: 10.1016/0167-6687(88)90091-1
[9]	E. S. W. Shiu, Calculation of the probability of eventual ruin by Beekman's convolution series, Insur. Math. Econ., 7 (1988), 41–47. https://doi.org/10.1016/0167-6687(88)90095-9 doi: 10.1016/0167-6687(88)90095-9
[10]	E. S. W. Shiu, Ruin probability by operational calculus, Insur. Math. Econ., 8 (1989), 243–249. https://doi.org/10.1016/0167-6687(89)90060-7. doi: 10.1016/0167-6687(89)90060-7
[11]	F. De Vylder, M. J. Goovaerts, Recursive calculation of finite-time ruin probabilities, Insur. Math. Econ., 7 (1988), 1–7. https://doi.org/10.1016/0167-6687(88)90089-3 doi: 10.1016/0167-6687(88)90089-3
[12]	P. Picard, C. Lefèvre, Probabilité de ruine éventuelle dans un modèle de risque à temps discret, J. Appl. Probab., 40 (2003), 543–556. https://doi.org/10.1239/jap/1059060887 doi: 10.1239/jap/1059060887
[13]	S. M. Li, F. J. Huang, C. Jin, Joint distributions of some ruin related quantities in the compound binomial risk model, Stoch. Models, 29 (2013), 518–539. https://doi.org/10.1080/15326349.2013.847610 doi: 10.1080/15326349.2013.847610
[14]	L. Rincón, D. J. Santana, Ruin probability for finite negative binomial mixture claims via recurrence sequences, Comm. Statist. Theory Methods, 2022. https://doi.org/10.1080/03610926.2022.2087091 doi: 10.1080/03610926.2022.2087091
[15]	Y. Q. Cang, Y. Yang, X. X. Shi, A note on the uniform asymptotic behavior of the finite-time ruin probability in a nonstandard renewal risk model, Lith. Math. J., 60 (2020), 161–172. https://doi.org/10.1007/s10986-020-09473-x doi: 10.1007/s10986-020-09473-x
[16]	C. Lefèvre, M. Simon, Schur-constant and related dependence models, with application to ruin probabilities, Methodol. Comput. Appl. Probab., 23 (2021), 317–339. https://doi.org/10.1007/s11009-019-09744-2 doi: 10.1007/s11009-019-09744-2
[17]	D. G. Kendall, The genealogy of genealogy branching processes before (and after) 1873, Bull. London Math. Soc., 7 (1975), 225–253. https://doi.org/10.1112/blms/7.3.225 doi: 10.1112/blms/7.3.225
[18]	L. Arguin, D. Belius, A. J. Harper, Maxima of a randomized Riemann zeta function, and branching random walks, Ann. Appl. Probab., 27 (2017), 178–215. https://doi.org/10.1214/16-AAP1201 doi: 10.1214/16-AAP1201
[19]	L. Arguin, D. Belius, P. Bourgade, M. Radziwiłł, K. Soundararajan, Maximum of the Riemann zeta function on a short interval of the critical line, Commun. Pure Appl. Math., 72 (2019), 500–535. https://doi.org/10.1002/cpa.21791 doi: 10.1002/cpa.21791
[20]	E. Rawashdeh, A simple method for finding the inverse matrix of Vandermonde matrix, Mat. Vesn., 71 (2019), 207–213.
[21]	W. Rudin, Real and complex analysis, 3 Eds., New York: McGraw-Hill, 1987.
[22]	A. Grigutis, J. Jaunkauskas, J. Šiaulys, Multi seasonal discrete time risk model revisited, 2022, In press. https://doi.org/10.48550/arXiv.2207.03196
[23]	R. A. Horn, C. R. Johnson, Topics in matrix analysis, Cambridge University Press, 1991. https://doi.org/10.1017/CBO9780511840371
[24]	J. Kiefer, J. Wolfowitz, On the characteristics of the general queueing process, with applications to random walk, Ann. Math. Statist., 27 (1956), 147–161. https://doi.org/10.1214/aoms/1177728354 doi: 10.1214/aoms/1177728354
[25]	Mathematica (Version 9.0), Champaign, Illinois: Wolfram Research, Inc., 2012. Available from: https://www.wolfram.com/mathematica.
[26]	O. Navickienė, J. Sprindys, J. Šiaulys, Ruin probability for the bi-seasonal discrete time risk model with dependent claims, Modern Stoch. Theory Appl., 6 (2018), 133–144. https://doi.org/10.15559/18-VMSTA118 doi: 10.15559/18-VMSTA118
[27]	Y. Miao, K. P. Sendova, B. L. Jones, On a risk model with dual seasonalities, N. Am. Actuar. J., 2022. https://doi.org/10.1080/10920277.2022.2068611 doi: 10.1080/10920277.2022.2068611
[28]	A. Alencenovič, A. Grigutis, Bi-seasonal discrete time risk model with income rate two, Commun. Statist. Theory Methods, 2022. https://doi.org/10.1080/03610926.2022.2026962 doi: 10.1080/03610926.2022.2026962
[29]	A. Grigutis, A. Nakliuda, Note on the bi-risk discrete time risk model with income rate two, Modern Stoch. Theory Appl., 9 (2022), 401–412. https://doi.org/10.15559/22-VMSTA209 doi: 10.15559/22-VMSTA209

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)