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Occupation times of refracted Lévy processes. (English) Zbl 1306.60049

Summary: A refracted Lévy process 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 precisely, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation \[ \mathrm dU_t=-\delta\mathbf 1 _{\{U_t>b\}}\mathrm dt +\mathrm dX_t,\quad t\geq 0, \] where \(X=(X_t, t\geq 0)\) is a Lévy process with law \(\mathbb P\) and \(b, \delta\in\mathbb R\) such that the resulting process \(U\) may visit the half line \((b,\infty)\) with positive probability. In this paper, we consider the case that \(X\) is spectrally negative and establish a number of identities for the following functionals \[ \int\limits_0^\infty\mathbf 1 _{\{U_t<b\}}\mathrm dt,\quad \int\limits_0^{\kappa_c^+}\mathbf 1 _{\{U_t<b\}}\mathrm dt,\quad \int\limits_0^{\kappa^-_a}\mathbf 1 _{\{U_t<b\}}\mathrm dt,\quad \int\limits_0^{\kappa_c^+\wedge\kappa^-_a}\mathbf 1 _{\{U_t<b\}}\mathrm dt, \] where \(\kappa^+_c=\inf\{t\geq 0: U_t> c\}\) and \(\kappa^-_a=\inf\{t\geq 0: U_t< a\}\) for \(a<b<c\). Our identities extend recent results of D. Landriault et al. [Stochastic Processes Appl. 121, No. 11, 2629–2641 (2011; Zbl 1227.60061)] and bear relevance to Parisian-type financial instruments and insurance scenarios.

MSC:

60G51 Processes with independent increments; Lévy processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)

Citations:

Zbl 1227.60061

References:

[1] Avram, F., Palmowski, Z., Pistorius, M.R.: On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Probab. 17, 156-180 (2007) · Zbl 1136.60032 · doi:10.1214/105051606000000709
[2] Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996) · Zbl 0861.60003
[3] Chow, Y.S., Teicher, H.: Independence Interchangeability Martingales. Springer, New Yok (1978) · Zbl 0399.60001
[4] Furrer, H.: Risk processes perturbed by \[\alpha\] α-stable Lévy motion. Scand. Actuar. J. 1998(1), 59-74 (1998) · Zbl 1026.60516
[5] Huzak, M., Perman, M., Šikić, H., Vondraček, Z.: Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Probab. 14, 1378-1397 (2004) · Zbl 1061.60075 · doi:10.1214/105051604000000332
[6] Huzak, M., Perman, M., Šikić, H., Vondraček, Z.: Ruin probabilities for competing claim processes. J. Appl. Probab. 41, 679-690 (2004) · Zbl 1065.60100 · doi:10.1239/jap/1091543418
[7] Kuznetsov, A., Kyprianou, A.E., Rivero, V.: The theory of scale functions for spectrally negative Lévy processes. Lévy Matters II, Springer Lecture Notes in Mathematics (2013) · Zbl 1261.60047
[8] Klüppelberg, C., Kyprianou, A.E., Maller, R.A.: Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab. 14, 1766-1801 (2004) · Zbl 1066.60049 · doi:10.1214/105051604000000927
[9] Klüppelberg, C., Kyprianou, A.E.: On extreme ruinous behaviour of Lévy insurance risk processes. J. Appl. Probab. 43(2), 594-598 (2006) · Zbl 1118.60071 · doi:10.1239/jap/1152413744
[10] Kyprianou, A.E.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin (2006) · Zbl 1104.60001
[11] Kyprianou, A.E., Loeffen, R.: Refracted Lévy processes. Ann. Inst. H. Poincaré 46(1), 24-44 (2010) · Zbl 1201.60042 · doi:10.1214/08-AIHP307
[12] Kyprianou, A.E., Palmowski, Z.: Distributional study of de Finetti’s dividend problem for a general Lévy insurance risk process. J. Appl. Probab. 44, 349-365 (2007) · Zbl 1137.60047 · doi:10.1239/jap/1183667412
[13] Landriault, D., Renaud, J-F., Zhou, X.: Insurance risk models with Parisian implementation delays, Submitted (2011) www.ssrn.com/abstract=1744193 · Zbl 1319.60098
[14] Landriault, D., Renaud, J.-F., Zhou, X.: Occupation times of spectrally negative Lévy processes with applications. Stoch. Process. Appl. 121, 2629-2641 (2011) · Zbl 1227.60061 · doi:10.1016/j.spa.2011.07.008
[15] Lambert, A., Simatos, F., Zwart, B.: Scaling limits via excursion theory: interplay between Crump-Mode-Jagers branching processes and processor-sharing queues. To appear in Ann. Appl. Prob. (2013) · Zbl 1285.60034
[16] Loeffen, R. L., Renaud, J-F., Zhou, X.: Occupation times of intervals untill first passage times for spectrally negative Lévy processes with applications. Submitted (2012) arxiv:1207.1592 · Zbl 1137.60047
[17] Renaud, J.-F., Zhou, X.: Distribution of the dividend payments in a general Lévy risk model. J. Appl. Probab. 44, 420-427 (2007) · Zbl 1132.60041 · doi:10.1239/jap/1183667411
[18] Song, R., Vondraček, Z.: On suprema of Lévy processes and application in risk theory. Ann. lnst. H. Poincaré 44, 977-986 (2008) · Zbl 1178.60036 · doi:10.1214/07-AIHP142
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