The Theory of Scale Functions for Spectrally Negative Lévy Processes
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- 作者：Alexey Kuznetsov (1)
Andreas E. Kyprianou (2)
Victor Rivero (3) - 关键词：Applied probability ; Excursion theory ; First passage problem ; Fluctuation theory ; Laplace transform ; Scale functions ; Spectrally negative Lévy processes
- 刊名：Lecture Notes in Mathematics
- 出版年：2013
- 出版时间：2013
- 年：2013
- 卷：1
- 期：1
- 页码：187-188
- 全文大小：952KB
- 参考文献：1. J. Abate, P.P. Valko, Multi-precision Laplace transform inversion. Int. J. Numer. Meth. Eng. 60, 979-93 (2004)
2. J. Abate, W. Whitt, The Fourier-series method for inverting transforms of probability distributions. Queueing Syst. 10, 5-8 (1992)
3. J. Abate, W. Whitt, A unified framework for numerically inverting Laplace transforms. INFORMS J. Comput. 18, 408-21 (2006)
4. J. Abate, G.L. Choudhury, W. Whitt, in / An Introduction to Numerical Transform Inversion and Its Application to Probability Models, ed. by W. Grassman. Computational Probability (Kluwer, Dordrecht, 1999), pp. 257-23
5. H-J. Albrecher, J-F. Renaud, X. Zhou, A Lévy insurance risk process with tax. J. Appl. Probab. 45, 363-75 (2008)
6. H-J. Albrecher, F. Avram, D. Kortschak, On the efficient evaluation of ruin probabilities for completely monotone claim distributions. J. Comput. Appl. Math. 233, 2724-736 (2010)
7. D. Applebaum, / Lévy Processes and Stochastic Calculus, 2nd edn. (Cambridge University Press, London, 2009)
8. S. Asmussen, F. Avram, M.R. Pistorius, Russian and American put options under exponential phase-type Lévy models. Stoch. Proc. Appl. 109, 79-11 (2004)
9. F. Avram, A.E. Kyprianou, M. Pistorius, Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Probab.?14, 215-38 (2004)
10. F. Avram, Z. Palmowski, M. Pistorius, On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Probab. 17, 156-80 (2007)
11. D.H. Bailey, A fortran 90-based multiprecision system. ACM Trans. Math. Software 21, 379-87 (1995)
12. D.H. Bailey, P.N. Swarztrauber, The fractional Fourier transform and applications. SIAM Rev. 33, 389-04 (1991)
13. C. Berg, G. Forst, in / Potential Theory on Locally Compact Abelian Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 87 (Springer, Berlin, 1975)
14. J. Bertoin, An extension of Pitman’s Theorem for spectrally positive Lévy processes. Ann. Probab. 20, 1464-483 (1992)
15. J. Bertoin, / Lévy Processes (Cambridge University Press, London, 1996)
16. J. Bertoin, On the first exit time of a completely asymmetric stable process from a finite interval. Bull. Lond. Math. Soc.?28, 514-20 (1996)
17. J. Bertoin, Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Probab.?7, 156-69 (1997)
18. K. Bichteler, / Stochastic Integration with Jumps (Cambridge University Press, London, 2002)
19. E. Biffis, A.E. Kyprianou, A note on scale functions and the time value of ruin for Lévy insurance risk processes. Insur. Math. Econ. 46, 85-1 (2010)
20. E. Biffis, M. Morales, On an extension of the Gerber-Shiu function to path-dependent penalties. Insur. Math. Econ. 46, 92-7 (2010)
21. N.H. Bingham, Fluctuation theory in continuous time. Adv. Appl. Probab.?7, 705-66 (1975)
22. N.H. Bingham, Continuous branching processes and spectral positivity. Stochast. Process. Appl. 4, 217-42 (1976)
23. L. Bondesson, in / Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Lecture Notes in Statistics, 76 (Springer, New York, 1992)
24. A.A. Borovkov, / Stochastic Processes in Queueing Theory (Springer, Berlin, 1976)
25. N.S. Bratiychuk, D.V. Gusak, / Boundary Problem for Processes with Independent Increments (Naukova Dumka, Kiev, 1991) (in Russian)
26. M.E. Caballero, A. Lambert, G. Uribe Bravo, Proof(s) of the Lamperti representation of continuous-state branching processes. Probab. Surv. 6, 62-9 (2009)
27. T. Chan, A.E. Kyprianou, M. Savov, Smoothness of scale functions for spectrally negative Lévy processes. Probab. Theor. Relat. Fields 150, 691-08 (2011)
28. L. Chaumont, Sur certains processus de Lévy conditionés a rester positifs. Stochast. Stochast. Rep. 47 (1994) 1-0 (1994)
29. L. Chaumont, Conditionings and path decompositions for Lévy processes. Stochast. Process. Appl. 64, 34-9 (1996)
30. L. Chaumont, R.A. Doney, On Lévy processes conditioned to stay positive. Electron. J. Probab. 10, 948-61 (2005)
31. L. Chaumont, A.E. Kyprianou, J.C. Pardo, Some explicit identities associated with positive self-similar Markov processes. Stochast. Process. Appl. 119(3), 980-000 (2009)
32. M. Chazal, A.E. Kyprianou, P. Patie, A transformation for Lévy processes with one-sided jumps and applications. Preprint (2010)
33. A.M. Cohen, / Numerical Methods for Laplace Transform Inversion, vol.?5 of Numerical Methods and Algorithms (Springer, Berlin, 2007)
34. R. Cont, P. Tankov, / Financial Modeling with Jump Processes (Chapman & Hall, London, 2004)
35. C. Donati-Martin, M. Yor, Further examples of explicit Krein representations of certain subordinators. / Publ. Res. Inst. Math. Sci. 43, 315-28 (2007)
36. R.A. Doney, in / Some Excursion Calculations for Spectrally One-Sided Lévy Processes. Séminaire de Probabilités, vol. XXXVIII. Lecture Notes in Math., 1857 (Springer, Berlin, 2005), pp. 5-5
37. L. D?ring, M. Savov, (Non)differentiability and asymptotics for renewal densities of subordinators. Electron. J. Probab. 16, 470-03 (2011)
38. P. Dube, F. Guillemin, R.R. Mazumdar, Scale functions of Lévy processes and busy periods of finite-capacity M/GI/1 queues. J. Appl. Probab. 41, 1145-156 (2004)
39. M. Egami, K. Yamazaki, On scale functions of spectrally negative Lévy processes with phase-type jumps. arXiv:1005.0064v3 [math.PR] (2010)
40. W. Feller, / An introduction to probability theory and its applications. Vol?II, 2nd edition (Wiley, New York, 1971)
41. L.N.G. Filon, On a quadrature formula for trigonometric integrals. Proc. R. Soc. Edinb. 49, 38-7 (1928)
42. B. de Finetti, Su un’impostazione alternativa dell teoria collecttiva del rischio. Transactions of the XVth International Congress of Actuaries, vol. 2, pp. 433-43 (1957)
43. L.D. Fosdick, A special case of the Filon quadrature formula. Math. Comp. 22, 77-1 (1968)
44. D.P. Gaver, Observing stochastic processes and approximate transform inversion. Oper. Res. 14, 444-59 (1966)
45. H.U. Gerber, E.S.W. Shiu, On the time value of ruin. North Am. Actuarial J. 2, 48-8 (1998)
46. P.E. Greenwood, J.W. Pitman, Fluctuation identities for Lévy processes and splitting at the maximum. Adv. Appl. Probab. 12 839-02 (1979)
47. J. Hawkes, On the potential theory of subordinators. Z. W. 33(2), 113-32 (1975)
48. J. Hawkes, Intersections of Markov random sets. Z. W. 37(3), 243-51 (1976)
49. F. Hubalek, A.E. Kyprianou, Old and new examples of scale functions for spectrally negative Lévy processes. / Sixth Seminar on Stochastic Analysis, Random Fields and Applications, eds R. Dalang, M. Dozzi, F. Russo. Progress in Probability, Birkhuser (2010) 119-46.
50. A. Iserles, On the numerical quadrature of highly-oscillating integrals I: Fourier transforms. IMA J. Numer. Anal. 24, 365-91 (2004)
51. N. Jacobs, / Pseudo Differential Operators and Markov Processes. Vol. I Fourier Analysis and Semigroups (Imperial College Press, London, 2001)
52. L.F. James, B. Roynette, M. Yor, Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples. Probab. Surv. 5, 346-15 (2008)
53. H. Kesten, in / Hitting Probabilities of Single Points for Processes with Stationary Independent Increments. Memoirs of the American Mathematical Society, No. 93 (American Mathematical Society, Providence, 1969)
54. J.F.C. Kingman, Markov transition probabilities. II. Completely monotonic functions. Z. W. 9, 1- (1967)
55. R. Knobloch, A.E. Kyprianou, Survival of homogenous fragmentation processes with killing (2011), http://arxiv.org/abs/1104.5078[math.PR]
56. V.S. Korolyuk, Boundary problems for a compound Poisson process. Theor. Probab. Appl. 19, 1-4 (1974)
57. V.S. Korolyuk, / Boundary Problems for Compound Poisson Processes (Naukova Dumka, Kiev, 1975) (in Russian)
58. V.S. Korolyuk, On ruin problem for compound Poisson process. Theor. Probab. Appl. 20, 374-76 (1975)
59. V.S. Korolyuk, Ju.V. Borovskich, / Analytic Problems of the Asymptotic Behaviour of Probability Distributions (Naukova Dumka, Kiev, 1981) (in Russian)
60. V.S. Korolyuk, V.N. Suprun, V.M. Shurenkov, Method of potential in boundary problems for processes with independent increments and jumps of the same sign. Theor. Probab. Appl. 21, 243-49 (1976)
61. N. Krell, Multifractal spectra and precise rates of decay in homogeneous fragmentations. Stochast. Proc. Appl. 118, 897-16 (2008)
62. A. Kuznetsov, Wiener-Hopf factorization and distribution of extrema for a family of Lévy processes. Ann. Appl. Probab. 20, 1801-830 (2010)
63. A. Kuznetsov, Wiener-Hopf factorization for a family of Lévy processes related to theta functions. J. Appl. Probab. 47, 1023-033 (2010)
64. A. Kuznetsov, M. Morales, Computing the finite-time expected discounted penalty function for a family of Lévy risk processes. Preprint (2010)
65. A. Kuznetsov, A.E. Kyprianou, J.C. Pardo, Meromorphic Lévy processes and their fluctuation identities. Ann. Appl. Probab. 22, 1101-135 (2012)
66. A.E. Kyprianou, / Introductory Lectures on Fluctuations of Lévy Processes and Their Applications (Springer, Berlin, 2006)
67. A.E. Kyprianou, J-C. Pardo, Continuous state branching processes and self-similarity. J.?Appl. Probab. 45, 1140-160 (2008)
68. A.E. Kyprianou, V. Rivero, Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electron. J. Probab. 13, 1672-701 (2008)
69. A.E. Kyprianou, X. Zhou, General tax structures and the Lévy insurance risk model. J. Appl. Probab. 46, 1146-156 (2009)
70. A.E. Kyprianou, V. Rivero, R. Song, Convexity and smoothness of scale functions and de Finetti’s control problem. J. Theor. Probab. 23, 547-64 (2010)
71. A.E. Kyprianou, R.L. Loeffen, J.L. Pérez, Optimal control with absolutely continuous strategies for spectrally negative Lévy processes. J. Appl. Probab. 49, 150-66 (2012)
72. A. Lambert, Completely asymmetric Lévy processes confined in a finite interval. Ann. Inst. H. Poincaré. Probab. Stat. 36, 251-74 (2000)
73. A. Lambert, Species abundance distributions in neutral models with immigra- tion or mutation and general lifetimes / J. Math. Biol. 63, 57-2 (2011)
74. A.L. Lewis, E. Mordecki, Wiener-hopf factorization for Lévy processes having positive jumps with rational transforms. J. Appl. Probab. 45, 118-34 (2008)
75. R.L. Loeffen, On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes. Ann. Appl. Probab. 18, 1669-680 (2008)
76. R.L. Loeffen, An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density. J. Appl. Probab. 46, 85-8 (2009)
77. R.L. Loeffen, An optimal dividends problem with transaction costs for spectrally negative Lévy processes. Insur. Math. Econ. 45, 41-8 (2009)
78. R.L. Loeffen, J-F. Renaud, De Finetti’s optimal dividends problem with an affine penalty function at ruin. Insur. Math. Econ. 46, 98-08 (2010)
79. E. Mordecki, The distribution of the maximum of a Lévy process with positive jumps of phase-type. Theor. Stochast. Proc. 8, 309-16 (2002)
80. P. Patie, Exponential functional of a new family of Lévy processes and self-similar continuous state branching processes with immigration. Bull. Sci. Math. 133, 355-82 (2009)
81. M.R. Pistorius, On doubly reflected completely asymmetric Lévy processes. Stochast. Proc. Appl. 107, 131-43 (2003)
82. M.R. Pistorius, On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theor. Probab. 17, 183-20 (2004)
83. M.R. Pistorius, in / A Potential-Theoretical Review of Some Exit Problems of Spectrally Negative Lévy Processes. Séminaire de Probabilités, vol. XXXVIII. Lecture Notes in Math., 1857 (Springer, Berlin, 2005), pp. 30-1
84. M. Pistorius, On maxima and ladder processes for a dense class of Lévy process. J. Appl. Probab. 43, 208-20 (2006)
85. J-F. Renaud, X. Zhou, Moments of the expected present value of total dividends until ruin in a Levy risk model. J. Appl. Probab. 44, 420-27 (2007)
86. D. Revuz, M. Yor, / Continuous Martingales and Brownian Motion, 3rd edn. (Springer, Berlin, 1999)
87. L.C.G. Rogers, Wiener-Hopf factorisation of diffusions and Lévy processes. Proc. Lond. Math. Soc. 47, 177-91 (1983)
88. L.C.G. Rogers, The two-sided exit problem for spectrally positive Lévy processes. Adv. Appl. Probab.?22, 486-87 (1990)
89. L.C.G. Rogers, Evaluating first-passage probabilities for spectrally one-sided Lévy processes. J. Appl. Probab. 37, 1173-180 (2000)
90. J. Rosiński, Tempering stable processes. Stochast. Proc. Appl. 117(6), 677-07 (2007)
91. K. Sato, / Lévy Processes and Infinitely Divisible Distributions (Cambridge University Press, London, 1999)
92. R.L. Schilling, R. Song, Z. Vondra?ek, in / Bernstein Functions. Theory and Applications. de?Gruyter Studies in Mathematics, vol. 37 (2010)
93. L. Shepp, A.N. Shiryaev, The Russian option: Reduced regret. Ann. Appl. Probab. 3, 603-31 (1993)
94. L. Shepp, A.N. Shiryaev, A new look at the pricing of the Russian option. Theor. Probab. Appl. 39, 103-20 (1994)
95. R. Song, Z. Vondra?ek, in / Potential Theory of Subordinate Brownian Motion, ed. by P.?Graczyk, A. Stos. Potential Analysis of Stable Processes and its Extensions, Springer Lecture Notes in Mathematics 1980 (Springer, Berlin, 2009), pp. 87-76
96. R. Song, Z. Vondra?ek, Some remarks on special subordinators. Rocky Mt. J. Math. 40, 321-37 (2010)
97. H. Stehfest, Algorithm 368: Numerical inversion of Laplace transforms. Comm. ACM 13, 47-9 (1970)
98. V.N. Suprun, Problem of destruction and resolvent of terminating processes with independent increments. Ukranian Math. J.?28, 39-5 (1976)
99. B.A. Surya, Evaluating scale functions of spectrally negative Lévy processes. J. Appl. Probab. 45, 135-49 (2008)
100. L. Takács, / Combinatorial Methods in the Theory of Stochastic Processes (Wiley, New York, 1967)
101. A. Talbot, The accurate numerical inversion of Laplace transforms. IMA J. Appl. Math. 23, 97-20 (1979)
102. O. Thorin, On the infinite divisibility of the lognormal distribution. Scand. Actuarial J. 3, 121-48 (1977)
103. M. Veillette, M. Taqqu, Numerical computation of first-passage times of increasing Lévy processes. Meth. Comput. Appl. Probab. 12, 695-29 (2010)
104. V. Vigon, Votre Lévy rampe-t-il? J. Lond. Math. Soc. 65, 243-56 (2002)
105. D.V. Widder, / The Laplace Transform (Princeton University Press, Princeton, 1941)
106. M. Winkel, Right inverses of non-symmetric Lévy processes. Ann. Probab. 30, 382-15 (2002)
107. V.M. Zolotarev, The first passage time of a level and the behaviour at infinity for a class of processes with independent increments. Theor.?Probab. Appl.?9, 653-61 (1964) - 作者单位：Alexey Kuznetsov (1)
Andreas E. Kyprianou (2)
Victor Rivero (3)
1. Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, Canada, M3J 1P3
2. Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA1 2UU, UK
3. Centro de Investigación en Matemáticas A.C., Calle Jalisco s/n, C.P. 36240, Guanajuato, Gto. Mexico - ISSN：1617-9692
文摘
The purpose of this review article is to give an up to date account of the theory and applications of scale functions for spectrally negative Lévy processes. Our review also includes the first extensive overview of how to work numerically with scale functions. Aside from being well acquainted with the general theory of probability, the reader is assumed to have some elementary knowledge of Lévy processes, in particular a reasonable understanding of the Lévy–Khintchine formula and its relationship to the Lévy–It? decomposition. We shall also touch on more general topics such as excursion theory and semi-martingale calculus. However, wherever possible, we shall try to focus on key ideas taking a selective stance on the technical details. For the reader who is less familiar with some of the mathematical theories and techniques which are used at various points in this review, we note that all the necessary technical background can be found in the following texts on Lévy processes; (Bertoin, Lévy Processes (1996); Sato, Lévy Processes and Infinitely Divisible Distributions (1999); Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes and Their Applications (2006); Doney, Fluctuation Theory for Lévy Processes (2007)), Applebaum Lévy Processes and Stochastic Calculus (2009).