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Refracted Lévy processes. (English) Zbl 1201.60042
The paper is devoted to the study of the dynamics of a refracted Lévy process, which is one-dimensional Lévy process, perturbed in a special way: a linear drift is subtracted from its increments whenever it exceeds a predetermined positive level. More formally, the authors consider a solution to the stochastic differential equation \[ U_t = X_t - \delta\int_0^t \mathbf{1}_{\{U_s>b >}ds,\quad t\geq 0, \] where \(X_t\) is a Lévy process, \(b\) is a predetermined level. The solution of this equation may be thought of as the aggregate of the insurance risk process, when the dividends are paid out at a rate \(\delta\) whenever it exceed the level \(b\).
The authors show that refracted Lévy processes exist as strong solutions to the above stochastic differential equation whenever \(X\) is a spectrally negative Lévy process. Further they investigate the dynamics of such processes and establish a suite of identities, written in terms of scale functions, related to one- and two-sided exit problems. Finally they cite the relevance of such identities in context of a number of applications of spectrally negative Lévy processes within the context of ruin probabilities.

60G51 Processes with independent increments; Lévy processes
91B70 Stochastic models in economics
93E20 Optimal stochastic control
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[1] S. Asmussen and M. Taksar. Controlled diffusion models for optimal dividend pay-out. Insurance Math. Econom. 20 (1997) 1-15. · Zbl 1065.91529 · doi:10.1016/S0167-6687(96)00017-0
[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. · Zbl 1136.60032 · doi:10.1214/105051606000000709
[3] R. Bekker, O. J. Boxma and J. A. C. Resing. Lévy processes with adaptable exponent. Preprint, 2007. · Zbl 1169.60022 · doi:10.1239/aap/1240319581
[4] J. Bertoin. Lévy Processes . Cambridge Univ. Press, Cambridge, 1996. · Zbl 0861.60003
[5] T. Chan and A. E. Kyprianou. Smoothness of scale functions for spectrally negative Lévy processes. Preprint, 2008.
[6] E. Eberlein and D. Madan. Short sale restrictions, rally fears and option markets. Preprint, 2008. · Zbl 1229.91303
[7] H. Furrer. Risk processes perturbed by \alpha -stable Lévy motion. Scand. Actuar. J. 1 (1998) 59-74. · Zbl 1026.60516 · doi:10.1080/03461238.1998.10413992
[8] H. Gerber and E. Shiu. On optimal dividends: From reflection to refraction. J. Comput. Appl. Math. 186 (2006) 4-22. · Zbl 1089.91023 · doi:10.1016/j.cam.2005.03.062
[9] H. Gerber and E. Shiu. On optimal dividend strategies in the compound Poisson model. N. Am. Actuar. J. 10 (2006) 76-93.
[10] B. Hilberink and L. C. G. Rogers. Optimal capital structure and endogenous default. Finance Stoch. 6 (2002) 237-263. · Zbl 1002.91019 · doi:10.1007/s007800100058
[11] F. Hubalek and A. E. Kyprianou. Old and new examples of scale functions for spectrally negative Lévy processes, 2007. Available at · Zbl 1274.60148
[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. · Zbl 1061.60075 · doi:10.1214/105051604000000332
[13] M. Huzak, M. Perman, H. Šikić and Z. Vondraček. Ruin probabilities for competing claim processes. J. Appl. Probab. 41 (2004) 679-690. · Zbl 1065.60100 · doi:10.1239/jap/1091543418
[14] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes . Springer, Berlin, 2003. · Zbl 1018.60002
[15] M. Jeanblanc and A. N. Shiryaev. Optimization of the flow of dividends. Uspekhi Mat. Nauk 50 (1995) 25-46. · Zbl 0878.90014
[16] I. Karatzas and S. Shreve. Brownian Motion and Stochastic Calculus , 2nd edition. Graduate Texts in Mathematics 113 . Springer, New York, 1991. · Zbl 0734.60060
[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. · Zbl 1066.60049 · doi:10.1214/105051604000000927 · euclid:aoap/1099674077
[18] C. Klüppelberg and A. E. Kyprianou. On extreme ruinous behaviour of Lévy insurance risk processes. J. Appl. Probab. 43 (2006) 594-598. · Zbl 1118.60071 · doi:10.1239/jap/1152413744
[19] A. E. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications . Springer, Berlin, 2006. · Zbl 1104.60001
[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. · Zbl 1137.60047 · doi:10.1239/jap/1183667412
[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. · Zbl 1193.60064 · emis:journals/EJP-ECP/_ejpecp/viewarticle12e5.html · eudml:230675
[22] A. E. Kyprianou, V. Rivero and R. Song. Convexity and smoothness of scale functions and de Finetti’s control problem, 2008. Available at · Zbl 1188.93115
[23] A. Lambert. Completely asymmetric Lévy processes confined in a finite interval. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) 251-274. · Zbl 0970.60055 · doi:10.1016/S0246-0203(00)00126-6 · numdam:AIHPB_2000__36_2_251_0 · eudml:77658
[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. · Zbl 1157.91383 · doi:10.1016/j.insmatheco.2005.08.001
[25] M. R. Pistorius. A potential theoretical review of some exit problems of spectrally negative Lévy processes. Séminaire de Probabilités 38 (2005) 30-41. · Zbl 1065.60047
[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. · Zbl 1132.60041 · doi:10.1239/jap/1183667411
[27] L. C. G. Rogers. Evaluating first-passage probabilities for spectrally one-sided Lévy processes. J. Appl. Probab. 37 (2000) 1173-1180. · Zbl 0981.60048 · doi:10.1239/jap/1014843099
[28] R. Situ. Theory of Stochastic Differential Equations with Jumps and Applications . Springer, New York, 2005. · Zbl 1070.60002 · doi:10.1007/b106901
[29] D. W. Strook. A Concise Introduction to the Theory of Integration , 3rd edition. Birkhäuser, Boston, 1999.
[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. · Zbl 1178.60036 · doi:10.1214/07-AIHP142 · eudml:78000
[31] B. A. Surya, Evaluating scale functions of spectrally negative Lévy processes. J. Appl. Probab. 45 (2008) 135-149. · Zbl 1140.60027 · doi:10.1239/jap/1208358957
[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. · Zbl 1183.91077 · doi:10.1016/j.insmatheco.2006.08.002
[33] A. Yu. Veretennikov. On strong solutions of stochastic. It equations with jumps. Teor. Veroyatnost. i Primenen. 32 (1987) 159-163. (In Russian.) · Zbl 0658.60087 · doi:10.1137/1132019
[34] W. Whitt. Stochastic-Process Limits . Springer, New York, 2002. · Zbl 0993.60001 · doi:10.1007/b97479
[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. · Zbl 1161.60334 · doi:10.1016/j.spl.2005.12.024
[36] X. Zhou. When does surplus reach a certain level before ruin? Insurance Math. Econom. 35 (2004) 553-561. · Zbl 1117.91387 · doi:10.1016/j.insmatheco.2004.07.012
[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.
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