Refracted Lévy processes

A. E. Kyprianou; R. L. Loeffen

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 1, page 24-44
  • ISSN: 0246-0203

Abstract

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Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Lévy processes. The latter is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation dUt=−δ1{Ut>b} dt+dXt, where X={Xt : t≥0} is a Lévy process with law ℙ and b, δ∈ℝ such that the resulting process U may visit the half line (b, ∞) with positive probability. We consider in particular the case that X is spectrally negative and establish a suite of identities for the case of one and two sided exit problems. All identities can be written in terms of the q-scale function of the driving Lévy process and its perturbed version describing motion above the level b. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes.

How to cite

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Kyprianou, A. E., and Loeffen, R. L.. "Refracted Lévy processes." Annales de l'I.H.P. Probabilités et statistiques 46.1 (2010): 24-44. <http://eudml.org/doc/240802>.

@article{Kyprianou2010,
abstract = {Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Lévy processes. The latter is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation dUt=−δ1\{Ut&gt;b\} dt+dXt, where X=\{Xt : t≥0\} is a Lévy process with law ℙ and b, δ∈ℝ such that the resulting process U may visit the half line (b, ∞) with positive probability. We consider in particular the case that X is spectrally negative and establish a suite of identities for the case of one and two sided exit problems. All identities can be written in terms of the q-scale function of the driving Lévy process and its perturbed version describing motion above the level b. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes.},
author = {Kyprianou, A. E., Loeffen, R. L.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stochastic control; fluctuation theory; Lévy processes},
language = {eng},
number = {1},
pages = {24-44},
publisher = {Gauthier-Villars},
title = {Refracted Lévy processes},
url = {http://eudml.org/doc/240802},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Kyprianou, A. E.
AU - Loeffen, R. L.
TI - Refracted Lévy processes
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 1
SP - 24
EP - 44
AB - Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Lévy processes. The latter is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation dUt=−δ1{Ut&gt;b} dt+dXt, where X={Xt : t≥0} is a Lévy process with law ℙ and b, δ∈ℝ such that the resulting process U may visit the half line (b, ∞) with positive probability. We consider in particular the case that X is spectrally negative and establish a suite of identities for the case of one and two sided exit problems. All identities can be written in terms of the q-scale function of the driving Lévy process and its perturbed version describing motion above the level b. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes.
LA - eng
KW - stochastic control; fluctuation theory; Lévy processes
UR - http://eudml.org/doc/240802
ER -

References

top
  1. [1] S. Asmussen and M. Taksar. Controlled diffusion models for optimal dividend pay-out. Insurance Math. Econom. 20 (1997) 1–15. Zbl1065.91529MR1466852
  2. [2] F. Avram, Z. Palmowski and M. R. Pistorius. On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Probab. 17 (2007) 156–180. Zbl1136.60032MR2292583
  3. [3] R. Bekker, O. J. Boxma and J. A. C. Resing. Lévy processes with adaptable exponent. Preprint, 2007. Zbl1169.60022MR2514950
  4. [4] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. Zbl0938.60005MR1406564
  5. [5] T. Chan and A. E. Kyprianou. Smoothness of scale functions for spectrally negative Lévy processes. Preprint, 2008. Zbl1259.60050
  6. [6] E. Eberlein and D. Madan. Short sale restrictions, rally fears and option markets. Preprint, 2008. Zbl1229.91303
  7. [7] H. Furrer. Risk processes perturbed by α-stable Lévy motion. Scand. Actuar. J. 1 (1998) 59–74. Zbl1026.60516MR1626676
  8. [8] H. Gerber and E. Shiu. On optimal dividends: From reflection to refraction. J. Comput. Appl. Math. 186 (2006) 4–22. Zbl1089.91023MR2190037
  9. [9] H. Gerber and E. Shiu. On optimal dividend strategies in the compound Poisson model. N. Am. Actuar. J. 10 (2006) 76–93. MR2328638
  10. [10] B. Hilberink and L. C. G. Rogers. Optimal capital structure and endogenous default. Finance Stoch. 6 (2002) 237–263. Zbl1002.91019MR1897961
  11. [11] F. Hubalek and A. E. Kyprianou. Old and new examples of scale functions for spectrally negative Lévy processes, 2007. Available at arXiv:0801.0393v1. Zbl1274.60148
  12. [12] M. Huzak, M. Perman, H. Šikić and Z. Vondraček. Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Probab. 14 (2004) 1378–1397. Zbl1061.60075MR2071427
  13. [13] M. Huzak, M. Perman, H. Šikić and Z. Vondraček. Ruin probabilities for competing claim processes. J. Appl. Probab. 41 (2004) 679–690. Zbl1065.60100MR2074816
  14. [14] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes. Springer, Berlin, 2003. Zbl1018.60002MR1943877
  15. [15] M. Jeanblanc and A. N. Shiryaev. Optimization of the flow of dividends. Uspekhi Mat. Nauk 50 (1995) 25–46. Zbl0878.90014MR1339263
  16. [16] I. Karatzas and S. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Graduate Texts in Mathematics 113. Springer, New York, 1991. Zbl0734.60060MR1121940
  17. [17] C. Klüppelberg, A. E. Kyprianou and R. A. Maller. Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab. 14 (2004) 1766–1801. Zbl1066.60049MR2099651
  18. [18] C. Klüppelberg and A. E. Kyprianou. On extreme ruinous behaviour of Lévy insurance risk processes. J. Appl. Probab. 43 (2006) 594–598. Zbl1118.60071MR2248586
  19. [19] A. E. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin, 2006. Zbl1104.60001MR2250061
  20. [20] A. E. Kyprianou and Z. Palmowski. Distributional study of de Finetti’s dividend problem for a general Lévy insurance risk process. J. Appl. Probab. 44 (2007) 349–365. Zbl1137.60047MR2340209
  21. [21] A. E. Kyprianou and V. Rivero. Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electron. J. Probab. 13 (2008) 1672–1701. Zbl1193.60064MR2448127
  22. [22] A. E. Kyprianou, V. Rivero and R. Song. Convexity and smoothness of scale functions and de Finetti’s control problem, 2008. Available at arXiv:0801.1951v2. MR2644875
  23. [23] A. Lambert. Completely asymmetric Lévy processes confined in a finite interval. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) 251–274. Zbl0970.60055MR1751660
  24. [24] X. S. Lin and K. P. Pavlova. The compound Poisson risk model with a threshold dividend strategy. Insurance Math. Econom. 38 (2006) 57–80. Zbl1157.91383MR2197303
  25. [25] M. R. Pistorius. A potential theoretical review of some exit problems of spectrally negative Lévy processes. Séminaire de Probabilités 38 (2005) 30–41. Zbl1065.60047MR2126965
  26. [26] J.-F. Renaud and X. Zhou. Distribution of the dividend payments in a general Lévy risk model. J. Appl. Probab. 44 (2007) 420–427. Zbl1132.60041MR2340208
  27. [27] L. C. G. Rogers. Evaluating first-passage probabilities for spectrally one-sided Lévy processes. J. Appl. Probab. 37 (2000) 1173–1180. Zbl0981.60048MR1808924
  28. [28] R. Situ. Theory of Stochastic Differential Equations with Jumps and Applications. Springer, New York, 2005. Zbl1070.60002MR2160585
  29. [29] D. W. Strook. A Concise Introduction to the Theory of Integration, 3rd edition. Birkhäuser, Boston, 1999. Zbl0912.28001MR1658777
  30. [30] R. Song and Z. Vondraček. On suprema of Lévy processes and application in risk theory. Ann. lnst. H. Poincaré Probab. Statist. 44 (2008) 977–986. Zbl1178.60036MR2453779
  31. [31] B. A. Surya, Evaluating scale functions of spectrally negative Lévy processes. J. Appl. Probab. 45 (2008) 135–149. Zbl1140.60027MR2409316
  32. [32] N. Wan. Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion. Insurance Math. Econom. 40 (2007) 509–523. Zbl1183.91077MR2311547
  33. [33] A. Yu. Veretennikov. On strong solutions of stochastic. It equations with jumps. Teor. Veroyatnost. i Primenen. 32 (1987) 159–163. (In Russian.) Zbl0623.60070MR890945
  34. [34] W. Whitt. Stochastic-Process Limits. Springer, New York, 2002. Zbl0993.60001MR1876437
  35. [35] H. Y. Zhang, M. Zhou and J. Y. Guo. The Gerber–Shiu discounted penalty function for classical risk model with a two-step premium rate. Statist. Probab. Lett. 76 (2006) 1211–1218. Zbl1161.60334MR2269347
  36. [36] X. Zhou. When does surplus reach a certain level before ruin? Insurance Math. Econom. 35 (2004) 553–561. Zbl1117.91387MR2106135
  37. [37] X. Zhou. Discussion on: On optimal dividend strategies in the compound Poisson model by H. Gerber and E. Shiu. N. Am. Actuar. J. 10 (2006) 79–84. MR2328656

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