Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Refracted Lévy processes

A. E. Kyprianou and R. L. Loeffen

Full-text: Open access

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>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.

Résumé

Motivé par des considérations classiques de la théorie du risque, nous étudions des problèmes de croisement de frontière par des processus de Lévy réfractés. Un processus de Lévy réfracté est un processus de Lévy dont la dynamique possède une derive linéaire fixe. Plus formellement, un processus de Lévy réfracté est décrit par l’unique solution forte, si elle existe, de l’équation différentielle stochastique

dUt=−δ1{Ut>b} dt+dXt,

X est un processus de Lévy et b, δ∈ℝ, de telle manière que le processus U peut visiter l’intervalle (b, ∞) avec probabilité positive. En particulier, nous considérons le cas où le processus de Lévy n’a pas de saut positif et nous fournissons des calculs explicites pour le problème de sortie d’un intervalle. Toutes les identités peuvent être écrites en terme de la fonction d’échelle associée au processus de Lévy sous-jacent et sa version perturbée. Finallement, nous appliquons les identités obtenues aux processus de risque.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 46, Number 1 (2010), 24-44.

Dates
First available in Project Euclid: 1 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1267454106

Digital Object Identifier
doi:10.1214/08-AIHP307

Mathematical Reviews number (MathSciNet)
MR2641768

Subjects
Primary: 60J99: None of the above, but in this section
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 91B70: Stochastic models

Keywords
Stochastic control Fluctuation theory Lévy processes

Citation

Kyprianou, A. E.; Loeffen, R. L. Refracted Lévy processes. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 1, 24--44. doi:10.1214/08-AIHP307. https://projecteuclid.org/euclid.aihp/1267454106.


Export citation

References

  • [1] S. Asmussen and M. Taksar. Controlled diffusion models for optimal dividend pay-out. Insurance Math. Econom. 20 (1997) 1–15.
  • [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.
  • [3] R. Bekker, O. J. Boxma and J. A. C. Resing. Lévy processes with adaptable exponent. Preprint, 2007.
  • [4] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996.
  • [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.
  • [7] H. Furrer. Risk processes perturbed by α-stable Lévy motion. Scand. Actuar. J. 1 (1998) 59–74.
  • [8] H. Gerber and E. Shiu. On optimal dividends: From reflection to refraction. J. Comput. Appl. Math. 186 (2006) 4–22.
  • [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.
  • [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.
  • [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.
  • [13] M. Huzak, M. Perman, H. Šikić and Z. Vondraček. Ruin probabilities for competing claim processes. J. Appl. Probab. 41 (2004) 679–690.
  • [14] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes. Springer, Berlin, 2003.
  • [15] M. Jeanblanc and A. N. Shiryaev. Optimization of the flow of dividends. Uspekhi Mat. Nauk 50 (1995) 25–46.
  • [16] I. Karatzas and S. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Graduate Texts in Mathematics 113. Springer, New York, 1991.
  • [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.
  • [18] C. Klüppelberg and A. E. Kyprianou. On extreme ruinous behaviour of Lévy insurance risk processes. J. Appl. Probab. 43 (2006) 594–598.
  • [19] A. E. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin, 2006.
  • [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.
  • [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.
  • [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.
  • [23] A. Lambert. Completely asymmetric Lévy processes confined in a finite interval. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) 251–274.
  • [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.
  • [25] M. R. Pistorius. A potential theoretical review of some exit problems of spectrally negative Lévy processes. Séminaire de Probabilités 38 (2005) 30–41.
  • [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.
  • [27] L. C. G. Rogers. Evaluating first-passage probabilities for spectrally one-sided Lévy processes. J. Appl. Probab. 37 (2000) 1173–1180.
  • [28] R. Situ. Theory of Stochastic Differential Equations with Jumps and Applications. Springer, New York, 2005.
  • [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.
  • [31] B. A. Surya, Evaluating scale functions of spectrally negative Lévy processes. J. Appl. Probab. 45 (2008) 135–149.
  • [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.
  • [33] A. Yu. Veretennikov. On strong solutions of stochastic. It equations with jumps. Teor. Veroyatnost. i Primenen. 32 (1987) 159–163. (In Russian.)
  • [34] W. Whitt. Stochastic-Process Limits. Springer, New York, 2002.
  • [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.
  • [36] X. Zhou. When does surplus reach a certain level before ruin? Insurance Math. Econom. 35 (2004) 553–561.
  • [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.