Exact joint laws associated with spectrally negative Lévy processes and applications to insurance risk theory

doi:10.1007/s11464-013-0186-5

Frontiers of Mathematics in China

2014, Vol. 9

Issue (6): 1453-1471 https://doi.org/10.1007/s11464-013-0186-5

本期目录

Exact joint laws associated with spectrally negative Lévy processes and applications to insurance risk theory

Chuancun YIN^1,^*(

),Kam C. YUEN²

1. School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
2. Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong, China

全文: PDF(175 KB)

Abstract：

We consider the spectrally negative Lévy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the minimum, the maximum, and the duration of negative values. We apply our results to insurance risk theory to find an explicit expression for the generalized expected discounted penalty function in terms of scale functions. Furthermore, a new expression for the generalized Dickson’s formula is provided.

Key words： Fluctuation identity spectrally negative Lévy processes suprema and infima generalized Dickson’s formula scale function occupation time

收稿日期: 2012-11-12 出版日期: 2014-10-29

Corresponding Author(s): Chuancun YIN

引用本文:

. [J]. Frontiers of Mathematics in China, 2014, 9(6): 1453-1471.
Chuancun YIN,Kam C. YUEN. Exact joint laws associated with spectrally negative Lévy processes and applications to insurance risk theory. Front. Math. China, 2014, 9(6): 1453-1471.

链接本文:

https://academic.hep.com.cn/fmc/CN/10.1007/s11464-013-0186-5
https://academic.hep.com.cn/fmc/CN/Y2014/V9/I6/1453

1	Alili L, Kyprianou A E. Some remarks on first passage of Lévy processes, the American put and pasting principles. Ann Appl Probab, 2005, 15: 2062-2080 https://doi.org/10.1214/105051605000000377
2	Asmussen S. Ruin Probabilities. Singapore: World Scientific, 2000
3	Avram F, Kyprianou A E, Pistorius M R. Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann Appl Probab, 2004, 14: 215-238 https://doi.org/10.1214/aoap/1075828052
4	Avram F, Palmowski Z, Pistorius M R. On the optimal dividend problem for a spectrally negative Lévy process. Ann Appl Probab, 2007, 17: 156-180 https://doi.org/10.1214/105051606000000709
5	Bertoin J. Lévy Processes. Cambridge Tracts in Mathematics, Vol 121. Cambridge: Cambridge University Press, 1996
6	Bertoin J. Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann Appl Probab, 1997, 7: 156-169 https://doi.org/10.1214/aoap/1034625257
7	Biffis E, Kyprianou A E. A note on scale functions and the time value of ruin for Lévy insurance risk processes. Insurance Math Econom, 2010, 46: 85-91 https://doi.org/10.1016/j.insmatheco.2009.04.005
8	Biffis E, Morales M. On a generalization of the Gerber-Shiu function to path-dependent penalties. Insurance Math Econom, 2010, 46: 92-97 https://doi.org/10.1016/j.insmatheco.2009.08.011
9	Bingham N H. Fluctuation theory in continuous time. Adv Appl Probab, 1975, 7: 705-766 https://doi.org/10.2307/1426397
10	Chaumont C, Kyprianou A, Pardo J. Some explicit identities associated with positive self-similar Markov processes. Stoch Proc Appl, 2009, 119(3): 980-1000 https://doi.org/10.1016/j.spa.2008.05.001
11	Chiu S N, Yin C C. Passage times for a spectrally negative Lévy process with applications to risk theory. Bernoulli, 2005, 11(3): 511-522 https://doi.org/10.3150/bj/1120591186
12	Doney R A. Fluctuation Theory for Lévy Processes. Lecture Notes in Mathematics, Vol 1897. Berlin: Springer, 2007
13	Doney R A, Kyprianou A E. Overshoots and undershoots of Lévy processes. Ann Appl Probab, 2006, 16(1): 91-106 https://doi.org/10.1214/105051605000000647
14	Dos Reis A D E. How long is the surplus below zero? Insurance Math Econom, 1993, 12: 23-38 https://doi.org/10.1016/0167-6687(93)90996-3
15	Emery D J. Exit problem for a spectrally positive process. Adv Appl Probab, 1973, 5: 498-520 https://doi.org/10.2307/1425831
16	Erder I, Klüppelberg C. The first passage event for sums of dependent Lévy processes with applications to insurance risk. Ann Appl Probab, 2009, 19(6): 2047-2079 https://doi.org/10.1214/09-AAP601
17	Garrido J, Morales M. On the expected discounted penalty function for Lévy risk processes. North American Actuar J, 2006, 10(4): 196-218 https://doi.org/10.1080/10920277.2006.10597421
18	Gerber H U, Shiu E S W. On the time value of ruin. North American Actuar J, 1998, 2(1): 48-78 https://doi.org/10.1080/10920277.1998.10595671
19	Hubalek F, Kyprianou A. Old and new examples of scale functions for spectrally negative Lévy processes. In: Dalang R, Dozzi M, Russo F, eds. Sixth Seminar on Stochastic Analysis, Random Fields and Applications. Progress in Probability. Boston: Birkh?user, 2010, 119-146
20	Huzak M M, Perman M, ?iki? H, Vondra?ek Z. Ruin probabilities and decompositions for general perturbed risk processes. Ann Appl Probab, 2006, 14(3): 1378-1397
21	Kadankov V F, Kadankova T V. On the distribution of duration of stay in an interval of the semi-continuous process with independent increments. Random Oper Stoch Equ, 2004, 12(4): 361-384 https://doi.org/10.1515/1569397042722355
22	Klüppelberg C, Kyprianou A E. On extreme ruinous behaviour of Lévy insurance risk processes. J Appl Probab, 2006, 43(2): 594-598 https://doi.org/10.1239/jap/1152413744
23	Klüppelberg C, Kyprianou A E, Maller R A. Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann Appl Probab, 2004, 14(4): 1766-1801 https://doi.org/10.1214/105051604000000927
24	Kyprianou A E. Introductory Lecture Notes on Fluctuations of Lévy Processes with Applications. Berlin: Springer-Verlag, 2006
25	Kyprianou A E, Palmowski Z. A martingale review of some fluctuation theory for spectrally negative Lévy processes. In: Séminaire de Probabilités XXXVIII. Lecture Notes in Math, Vol 1857. Berlin: Springer, 2005, 16-29
26	Kyprianou A E, Palmowski Z. Distributional study of De Finetti’s dividend problem for a general Lévy insurance risk process. J Appl Probab, 2007, 44: 428-443 https://doi.org/10.1239/jap/1183667412
27	Kyprianou A E, Pardo J C, Rivero V. Exact and asymptotic n-tuple laws at first and last passage. Ann Appl Probab, 2010, 20(2): 522-564 https://doi.org/10.1214/09-AAP626
28	Kyprianou A E, Rivero V, Song R. Convexity and smoothness of scale functions and De Finetti’s control problem. J Theor Probab, 2010, 23: 547-564 https://doi.org/10.1007/s10959-009-0220-z
29	Landriault D, Renaud J, Zhou X W. Occupation times of spectrally negative Lévy processes with applications. Stochastic Process Appl, 2011, 121(11): 2629-2641 https://doi.org/10.1016/j.spa.2011.07.008
30	Loeffen R. On optimality of the barrier strategy in de Finetti’ dividend problem for spectrally negative Lévy processes. Ann Appl Probab, 2009, 18(5): 1669-1680 https://doi.org/10.1214/07-AAP504
31	Morales M. On the expected discounted penalty function for a perturbed risk process driven by a subordinator. Insurance Math Econom, 2007, 40(2): 293-301 https://doi.org/10.1016/j.insmatheco.2006.04.008
32	Pistorius M R. A potential-theoretical review of some exit problems of spectrally negative Lévy processes. In: Séminaire de Probabilités XXXVIII. Lecture Notes in Math, Vol 1857. Berlin: Springer, 2005, 30-41
33	Renaud J F, Zhou X. Distribution of the present value of dividend payments in a Lévy risk model. J Appl Probab, 2007, 44(2): 420-427 https://doi.org/10.1239/jap/1183667411
34	Rolski T, Schmidli H, Schmidt V, Teugels J. Stochastic Processes for Insurance and Finance. Chichester: Wiley, 1999 https://doi.org/10.1002/9780470317044
35	Yang H L, Zhang L Z. Spectrally negative Lévy processes with applications in risk theory. Adv Appl Probab, 2001, 33(1): 281-291 https://doi.org/10.1239/aap/999187908
36	Zhou X W. Some fluctuation identities for Lévy processes with jumps of the same sign. J Appl Probab, 2004, 41: 1191-1198 https://doi.org/10.1239/jap/1101840564
37	Zhou X W. On a classical risk model with a constant dividend barrier. North American Actuar J, 2005, 9: 95-108 https://doi.org/10.1080/10920277.2005.10596228
38	Zhang C S, Wang G J. The joint density function of three characteristics on jumpdiffusion risk process. Insurance Math Econom, 2003, 32: 445-455 https://doi.org/10.1016/S0167-6687(03)00133-1
39	Zhang C S, Wu R. Total duration of negative surplus for the compound Poisson process that is perturbed by diffusion. J Appl Probab, 2002, 39: 517-532 https://doi.org/10.1239/jap/1034082124

Viewed

Full text

Abstract

Cited

Shared

Discussed