Abstract
We consider an insurance risk model with extended flexibility, under which claims arrive according to a point process with an order statistics (OS) property, their amounts may have any joint distribution and the premium income is accumulated following any non-decreasing, possibly discontinuous real valued function. We generalize the definition of an OS point process, assuming it is generated by an arbitrary cdf allowing jump discontinuities, which corresponds to an arbitrary (possibly discontinuous) claim arrival cumulative intensity function. The latter feature is appealing for insurance applications since it allows to consider clusters of claims arriving instantaneously. Under these general assumptions, a closed form expression for the joint distribution of the time to ruin and the deficit at ruin is derived, which remarkably involves classical Appell polynomials. Corollaries of our main result generalize previous non-ruin formulas e.g., those obtained by Ignatov and Kaishev (Scand Actuar J 2000(1):46–62, 2000; J Appl Probab 41(2):570–578, 2004; J Appl Probab 43:535–551, 2006) and Lefèvre and Loisel (Methodol Comput Appl Probab 11(3):425–441, 2009) for the case of stationary Poisson claim arrivals and by Lefèvre and Picard (Insurance Math Econom 49:512–519, 2011; Methodol Comput Appl Probab 16:885–905, 2014), for OS claim arrivals.
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References
Aurzada F, Doering L, Savov M (2013) Small time Chung-type LIL for Lévy processes. Bernoulli 19(1):115–136
Berg M, Spizzichino F (2000) Time-lagged point processes with the order-statistics property. Math Meth Oper Res 2000(51):301–314
Bernyk V, Dalang RC, Peskir G (2008) The law of the supremum of stable Lévy processes with no negative jumps. Ann Probab 36:1777–1789
Bertoin J, Doney RA, Maller RA (2008) Passage of Lévy processes across power law boundaries at small times. Ann Probab 36(1):160–197
Borovkov AA (1964) On the first passage time for one class of processes with independent increments. Theor Probab Appl 10:331–334
Boucher JP , Denuit M, Guillén M (2007) Risk classification for claim counts: A comparative analysis of various zero inflated mixed Poisson and hurdle models. North American Actuarial Journal 11(4):110–131
Bühlmann H (1970) Mathematical Methods in Risk theory. Springer, Heidelberg
Crump KS (1975) On point processes having an order statistics structure. Sankhyā:The Indian Journal of Statistics Series A 37(3):396–404
Denuit M, Maréchal X, Pitrebois S, Walhin J-F (2007) Actuarial modelling of claim counts: Risk classification, credibility and bonus-malus systems. Wiley, West Sussex
De Vylder FF, Goovaerts MJ (1999a) Inequality extensions of Prabhu’s formula in ruin theory. Insurance Math Econom 24:249–271
De Vylder FF, Goovaerts MJ (1999b) Homogeneous risk models with equalized claim amounts. Insurance Math Econom 26:223–238
Dimitrova DS, Ignatov ZG, Kaishev VK (2014) Ruin and deficit under claim arrivals with the order statistics property. preprint. Available at:. https://www.researchgate.net/publication/306364689_Ruin_and_deficit_under_claim_arrivals_with_the_order_statistics_property
Dimitrova DS, Kaishev VK, Zhao S (2016) On the evaluation of finite-time ruin probabilities in a dependent risk model. Appl Math Comput 275:268–286
Doney RA (1991) Hitting probabilities for spectrally positive Lévy processes. J London Math Soc 44(3):556–576
Doney RA, Kyprianou AE (2006) Overshoots and undershoots of Lévy process. Ann Appl Probab 16:91–106
Eder I, Klüppelberg C (2009) The first passage event for sums of dependent Lévy processes with applications to insurance risk. Ann Appl Probab 19(6):2047–2079
Feigin PD (1979) On the characterization of point processes with the order statistic property. J Appl Probab 16:297–304
Garrido J, Morales M (2006) On The Expected Discounted Penalty function for Lévy Risk Processes. North American Actuarial Journal 10(4):196–216
Gerber HU, Shiu ESW (1997) The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance Math Econom 21:129–137
Gerber HU, Shiu ESW (1998) On the time value of ruin. North American Actuarial Journal 2:48–78
Haccou P, Jagers P, Vatutin VA (2005) Branching processes: Variation, growth, and extinction of populations. Cambridge University Press, Cambridge
Holmes PT (1971) On a property of Poisson process. Sankhyā:The Indian Journal of Statistics Series A 33(1):93–98
Huang WJ, Shoung JM (1994) On a study of some properties of point processes. Sankhyā:The Indian Journal of Statistics Series A 56(1):67–76
Huzak M, Perman M, Sikic H, Vondracek Z (2004) Ruin probabilities and decompositions for general perturbed risk processes. Ann Appl Probab 14(3):1378–1397
Ignatov ZG, Kaishev VK (2000) Two-sided Bounds for the Finite-time Probability of Ruin. Scand Actuar J 2000(1):46–62
Ignatov ZG, Kaishev VK (2004) A finite-time ruin probability formula for continuous claim severites. J Appl Probab 41(2):570–578
Ignatov ZG, Kaishev VK (2006) On the infinite-horizon probability of (non)ruin for integer-valued claims. J Appl Probab 43:535–551
Ignatov ZG, Kaishev VK (2016) First crossing time, overshoot and Appell-Hessenberg type functions. Stochastics 88(8):1240–1260
Ignatov ZG, Kaishev VK, Krachunov RS (2001) An Improved Finite-time Ruin Probability Formula and its ”Mathematica” Implementation. Insurance Math Econom 29(3):375–386
Kallenberg O (1976) Random measures. Akademie-Verlag, Berlin
Kendall D (1949) Stochastic Processes and Population Growth. J R Stat Soc Ser B 11(2):230–282
Kou SG, Wang H (2003) First passage times of a jump diffusion process. Adv Appl Probab 35:504–531
Klüppelberg C, Kyprianou AE, Maller RA (2004) Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann Appl Probab 14(4):1766–1801
Landriault D, Willmot GE (2009) On the joint distributions of the time to ruin, the surplus prior to ruin, and the deficit at ruin in the classical risk model. North American Actuarial Journal 13:252–270
Lefèvre C., Loisel S (2009) Finite-time ruin probabilities for discrete, possibly dependent, claim severities. Methodol Comput Appl Probab 11(3):425–441
Lefèvre C, Picard P (2011) A new look at the homogeneous risk model. Insurance Math Econom 49:512–519
Lefèvre C, Picard P (2014) Ruin Probabilities for Risk Models with Ordered Claim Arrivals. Methodol Comput Appl Probab 16:885–905
Liberman U (1985) An order statistic characterization of the Poisson renewal process. J Appl Probab 22:717–722
Nawrotzki K (1962) Ein Grenzwertsatz fr homogene zufllige Punktfolgen (Verallgemeinerung eines Satzes von A. Rényi). Math Nachricht 24:201–217
Peskir G (2007) The law of the passage times to points by a stable Lévy process with no-negative jumps. Research Report No 15, Probability and Statistics Group School of Mathematics, The Univ. Manchester
Picard P, Lefèvre C (1997) The probability of ruin in finite time with discrete claim size distribution. Insurance Math Econom 20(3):260–261
Puri PS (1982) On the characterization of point processes with the order statistic property without the moment condition. J Appl Probab 19:39–51
Savov M (2009) Small time two-sided LIL behavior for Lévy processes at zero. Probab Theory and Related Fields 144(1-2):79–98
Sendova KP, Zitikis R (2012) The order statistic claim process with dependent claim frequencies and severities. J Stat Theory Practice 6(4):597–620
Shi P, Valdez E (2014) Longitudinal modeling of insurance claim counts using jitters. Scandinavian Actuarial Journal 2014(2):159–179
Vein R, Dale P (1999) Determinants and their applications in mathematical physics. Springer-Verlag, New York
Westcott M (1973) Some remarks on a property of the Poisson process. Sankhyā:The Indian Journal of Statistics Series A 35:29–34
Willmot GE (1989) The total claims distribution under inflationary conditions. Scand Actuar J 1:1–12
Yang H, Zhang L (2001) Spectrally negative Levy processes with applications in risk theory. Adv Appl Probab 33(1):28191
Zolotarev VM (1964) The first passage time of a level and the behavior at infinity for a class of processes with independent increments. Theor Probab Appl 9:653–664
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Dimitrova, D.S., Ignatov, Z.G. & Kaishev, V.K. Ruin and Deficit Under Claim Arrivals with the Order Statistics Property. Methodol Comput Appl Probab 21, 511–530 (2019). https://doi.org/10.1007/s11009-018-9669-5
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DOI: https://doi.org/10.1007/s11009-018-9669-5
Keywords
- Order statistics point process
- Appell polynomials
- Hessenberg determinants
- Risk process
- Ruin probability
- First crossing time
- Overshoot