# zbMATH — the first resource for mathematics

Discrete time ruin probability with Parisian delay. (English) Zbl 1402.91188
Summary: In this paper we evaluate the probability of the discrete time Parisian ruin that occurs when surplus process stays below or at zero at least for some fixed duration of time $$d>0$$. We identify expressions for the ruin probabilities within finite and infinite-time horizon. We also find their light and heavy-tailed asymptotics when initial reserves approach infinity. Finally, we calculate these probabilities for a few explicit examples.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 60K10 Applications of renewal theory (reliability, demand theory, etc.) 60G51 Processes with independent increments; Lévy processes 62P05 Applications of statistics to actuarial sciences and financial mathematics
Full Text:
##### References:
 [1] Albrecher, H., Kortschak, D. & Zhou, X. (2012). Pricing of Parisian options for a jump-diffusion model with two-sided jumps. Applied Mathematical Finance 19(2), 97-129. · Zbl 1372.91100 [2] Alili, L., Chaumont, L. & Doney, R. (2005). On a fluctuation identity for random walks and Lévy processes. Bulletin of the London Mathematical Society 37, 141-148. · Zbl 1063.60062 [3] Asmussen, S. (2003). Applied probability and queues. New York: Springer. · Zbl 1029.60001 [4] Asmussen, S. & Albrecher, H. (2010). Ruin probabilities. Singapore: World Scientific. · Zbl 1247.91080 [5] Baurdoux, E. J., Pardo, J. C., Pérez, J. L. & Renaud, J.-F. (2016). Gerber-Shiu functionals at Parisian ruin for Lévy insurance risk processes. Advances in Applied Probability53, 572-584. · Zbl 1344.60046 [6] Bertoin, J. (1996). Lévy processes. Cambridge: Cambridge University Press. [7] Bertoin, J. & Doney, R. (1996). Some Asymptotic results for transient random walks. Advances in Applied Probability 28(1), 207-226. · Zbl 0854.60069 [8] Cheng, S., Gerber, H. U. & Shiu, E. S. W. (2000). Discounted probabilities and ruin theory in the compound binomial model. Insurance: Mathematics and Economics 26, 239-250. · Zbl 1013.91063 [9] Chesney, M., Jeanblanc-Picqué, M. & Yor, M. (1997). Brownian excursions and Parisian barrier options. Advances in Applied Probability29(1), 165-184. · Zbl 0882.60042 [10] Cossette, H., Landriault, D. & Marceau, E. (2003). Ruin probabilities in the compound Markov binomial model. Scandinavian Actuarial Journal 4, 301-323. · Zbl 1092.91040 [11] Cossette, H., Landriault, D. & Marceau, E. (2004). Compound binomial risk model in a Markovian environment. Insurance: Mathematics and Economics 35, 425-443. · Zbl 1079.91049 [12] Cossette, H., Landriault, D. & Marceau, E. (2006). Ruin probabilities in the discrete time renewal risk model. Insurance: Mathematics and Economics 38, 309-323. · Zbl 1090.60076 [13] Czarna, I. & Palmowski, Z. (2011). Ruin probability with Parisian delay for a spectrally negative Lévy risk process. Journal of Applied Probability 48(4), 984-1002. · Zbl 1232.60036 [14] Dassios, A. & Wu, S. (2009a). Parisian ruin with exponential claims. Submitted for publication. http://stats.lse.ac.uk/angelos/ [15] Dassios, A. & Wu, S. (2009b). Ruin probabilities of the Parisian type for small claims. Submitted for publication. http://stats.lse.ac.uk/angelos/ [16] De Vylder, F. E. & Goovaerts, M. J. (1988). Recursive calculation of finite-time ruin probabilities. Insurance: Mathematics and Economics 7, 1-7. · Zbl 0629.62101 [17] Dickson, D. C. M. (1992). On the distribution of surplus prior to ruin. Insurance: Mathematics and Economics 11, 191-207. · Zbl 0770.62090 [18] Dickson, D. C. M. (1994). Some comments on the compound binomial model. ASTIN Bulletin 24, 33-45. [19] Dickson, D. C. M. (2005). Insurance risk and ruin. Cambridge: Cambridge University Press. · Zbl 1060.91078 [20] Dickson, D. C. M., dos Reis, E. A. D. & Waters, H. R. (1995). Some stable algorithms in ruin theory and their applications. ASTIN Bulletin 25, 153-175. [21] Dickson, D. C. M. & Hipp, C. (2001). On the time to ruin for Erlang(2) risk process. Insurance: Mathematics and Economics 29, 333-344. · Zbl 1074.91549 [22] Dickson, D. C. M. & Waters, H. R. (1991). Recursive calculation of survival probabilities. ASTIN Bulletin 21, 199-221. [23] dos Reis, A. E. (1993). The compound binomial model revisited. Manuscript. Available at http://www.actuaries.org/ASTIN/Colloquia/Bergen/EgidiodosReis.pdf [24] Feller, W. (1966). An introduction to probability theory and its applications, Vol. II. New York: Wiley. · Zbl 0138.10207 [25] Foss, S., Korshunov, D. & Zachary, S. (2011). An introduction to heavy-tailed and subexponential distributions. New York: Springer. · Zbl 1250.62025 [26] Gerber, E. (1988). Mathematical fun with the compound binomial process. ASTIN Bulletin 18, 161-168. [27] Gerber, H. U. & Shiu, E. S. W. (1998). On the time value of ruin. North American Actuarial Journal 2(1), 48-78. · Zbl 1081.60550 [28] Gerber, H. U. & Shiu, E. S. W. (2005). The time value of ruin in a Sparre Andersen model. North American Actuarial Journal 9(2), 49-69. · Zbl 1085.62508 [29] Klüppelberg, C. & Kyprianou, A. (2006). On extreme ruinous behaviour of Lévy insurance risk process. Journal of Applied Probability 43(2), 594-598. · Zbl 1118.60071 [30] Landriault, D. (2008). On a generalization of the expected discounted penalty function in a discrete-time insurance risk model. Applied Stochastic Models in Business and Industry 24, 525-539. · Zbl 1199.91084 [31] Landriault, D., Renaud, J. F. & Zhou, X. (2014). Insurance risk models with Parisian implementation delays. Methodology and Computing in Applied Probability 16(3), 583-607. · Zbl 1319.60098 [32] Lefévre, C. & Loisel, S. (2008). On finite-time ruin probabilities for classisal risk models. Scandinavian Actuarial Journal 1, 41-60. [33] Li, S. (2005). On a class of discrete-time renewal risk models. Scandinavian Actuarial Journal 4, 241-260. · Zbl 1142.91043 [34] Li, S. (2005). Distributions of the surplus before ruin, the deficit at ruin and the claim causing ruin in a class of discrete time risk models. Scandinavian Actuarial Journal 4, 271-284. · Zbl 1143.91033 [35] Li, S. & Garrido, J., On the time value of ruin in the discrete time risk model. Working paper 02-18, Business Economics. Madrid: University Carlos III of Madrid, 2002. [36] Li, S. & Garrido, J. (2004). On ruin for Erlang(n) risk process. Insurance: Mathematics and Economics 34, 391-408. · Zbl 1188.91089 [37] Li, S. & Garrido, J. (2005). On a general class of renewal risk process: analysis of the Gerber-Shiu penalty function. Advances in Applied Probability 37, 836-856. · Zbl 1077.60063 [38] Li, S., Lu, Y. & Garrido, J. (2009). A review of discrete-time risk models. R ACSAM - Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales. Serie A. Matematicas103(2), 321-337. · Zbl 1180.62151 [39] Lin, X. S. & Willmot, G. E. (1999). Analysis of a defective renewal arising in ruin theory. Insurance: Mathematics and Economics 25, 63-84. · Zbl 1028.91556 [40] Lin, X. S. & Willmot, G. E. (2000). The moments of the time of ruin, the surplus before ruin and the deficit at ruin. Insurance: Mathematics and Economics 27, 19-44. · Zbl 0971.91031 [41] Liu, S. X. & Guo, J. Y. (2006). Discrete risk model revisited. Methodology and Computing in Applied Probability 8(2), 303-313. · Zbl 1098.91074 [42] Loeffen, R., Czarna, I. & Palmowski, Z. (2013). Parisian ruin probability for spectrally negative Lévy processes. Bernoulli 19(2), 599-609. · Zbl 1267.60054 [43] Michel, R. (1989). Representation of a time-discrete probability of eventual ruin. Insurance: Mathematics and Economics 8, 149-152. · Zbl 0676.62085 [44] Pavlova, K. & Willmot, G. E. (2004). The discrete stationary renewal risk model and the Gerber-Shiu discounted penalty function. Insurance: Mathematics and Economics 35, 267-277. · Zbl 1103.91046 [45] Rolski, T., Schmidli, H., Schmidt, V. & Teugels, J. L. (1999). Stochastic processes for insurance and finance. New York: John Wiley and Sons. · Zbl 0940.60005 [46] Shiu, E. (1989). The probability of eventual ruin in the compound binomial model. ASTIN Bulletin 19, 179-190. [47] Tang, Q. (2006). On convolution equivalence with applications. Bernoulli 12(3), 535-549. · Zbl 1114.60015 [48] Tang, Q. & Tsitsiashvili, G. (2003). Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stochastic Processes and Their Applications 108(2), 299-325. · Zbl 1075.91563 [49] Willmot, G. E. (1993). Ruin probabilities in the compound binomial model. Insurance: Mathematics and Economics 12, 133-142. · Zbl 0778.62099 [50] Willmot, G. E. (1999). A Laplace transform representation in a class of renewal queueing and risk processes. Journal of Applied Probability 36, 570-584. · Zbl 0942.60086 [51] Willmot, G. E. & Lin, X. S. (2001). Lundberg approximations for compound distributions with insurance applications. Lecture notes in statistics. New York: Springer-Verlag. [52] Wu, X. & Li, S. (2009). On the Gerber-Shiu function in a discrete time renewal risk model with general inter-claim times. Scandinavian Actuarial Journal 4, 281-294. · Zbl 1224.91094 [53] Yang, H., Zhang, Z. & Lan, C. (2009). Ruin problems in a discrete Markov risk model. Statistics and Probability Letters 79, 21-28. · Zbl 1153.62084 [54] Yuen, K. C. & Guo, J. (2001). Ruin probabilities for time-correlated claims in the compound binomial model. Insurance: Mathematics and Economics 29, 47-57. · Zbl 1074.91032 [55] Yuen, K. C. & Guo, J. (2006). Some results on the compound binomial model. Scandinavian Actuarial Journal 3, 129-140. · Zbl 1144.91036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.