Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T11:34:28.208Z Has data issue: false hasContentIssue false
  • English
  • Français

General drawdown of general tax model in a time-homogeneous Markov framework

Part of: Stochastic processes

Published online by Cambridge University Press:  22 November 2021

Florin Avram ,
Bin Li  and
Shu Li
Florin Avram*
Affiliation:
Université de Pau
Bin Li*
Affiliation:
University of Waterloo
Shu Li*
Affiliation:
Western University
*
*Postal address: Laboratoire de Mathématiques Appliquées, Université de Pau, 64013 Pau Cedex, France. Email address: florin.avram@univ-Pau.fr
**Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada. Email address: bin.li@uwaterloo.ca
***Postal address: Department of Statistical and Actuarial Sciences, Western University, London, ON, N6A 5B7, Canada. Email address: shu.li@uwo.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

Drawdown/regret times feature prominently in optimal stopping problems, in statistics (CUSUM procedure), and in mathematical finance (Russian options). Recently it was discovered that a first passage theory with more general drawdown times, which generalize classic ruin times, may be explicitly developed for spectrally negative Lévy processes [9, 20]. In this paper we further examine the general drawdown-related quantities in the (upward skip-free) time-homogeneous Markov process, and then in its (general) tax process by noticing the pathwise connection between general drawdown and the tax process.

Keywords

General drawdowntax processtime-homogeneous Markov process

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60G17: Sample path properties
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 4 , December 2021 , pp. 1131 - 1151
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albrecher, H. and Asmussen, S. (2010). Ruin Probabilities, 2nd edn. World Scientific, Singapore.Google Scholar
Albrecher, H. and Hipp, C. (2007). Lundberg’s risk process with tax. Blätter der DGVFM 28, 13–28.CrossRefGoogle Scholar
Albrecher, H. and Ivanovs, J. (2014). Power identities for Lévy risk models under taxation and capital injections. Stoch. Syst. 4, 157–172.10.1287/12-SSY079CrossRefGoogle Scholar
Albrecher, H., Avram, F., Constantinescu, C. and Ivanovs, J. (2014). The tax identity for Markov additive risk processes. Methodology Comput. Appl. Prob. 16, 245–258.CrossRefGoogle Scholar
Albrecher, H., Borst, S., Boxma, O. and Resing, J. (2009). The tax identity in risk theory: a simple proof and an extension. Insurance Math. Econom. 44, 304–306.10.1016/j.insmatheco.2008.05.001CrossRefGoogle Scholar
Albrecher, H., Renaud, J. F. and Zhou, X. (2008). A Lévy insurance risk process with tax. J. Appl. Prob. 45, 363–375.10.1239/jap/1214950353CrossRefGoogle Scholar
Avram, F., Grahovac, D. and Vardar-Acar, C. (2020). The W, Z scale functions kit for first passage problems of spectrally negative Levy processes, and applications to control problems. ESAIM: Prob. Statist. 24, 454–525.10.1051/ps/2019022CrossRefGoogle Scholar
Avram, F., Kyprianou, A. and Pistorius, M. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Prob. 14, 215–238.CrossRefGoogle Scholar
Avram, F., Vu, N. and Zhou, X. (2017). On taxed spectrally negative Lévy processes with draw-down stopping. Insurance Math. Econom. 76, 69–74.10.1016/j.insmatheco.2017.06.005CrossRefGoogle Scholar
Azéma, J. and Yor, M. (1979). Une solution simple au problème de Skorokhod. In Séminaire de Probabilités (Strasbourg) XIII, pp. 90–115.CrossRefGoogle Scholar
Bertoin, J. (1998). Lévy Processes. Cambridge University Press.Google Scholar
Carr, P. (2014). First-order calculus and option pricing. J. Financial Eng. 1, 1450009.10.1142/S2345768614500093CrossRefGoogle Scholar
Czarna, I., Pérez, J., Rolski, T. and Yamazaki, K. (2019). Fluctuation theory for level-dependent Lévy risk process. Stoch. Process. Appl. 129, 5406–5449.CrossRefGoogle Scholar
Kyprianou, A. (2014). Fluctuations of Lévy Processes with Applications: Introductory Lectures. Springer, Heidelberg.10.1007/978-3-642-37632-0CrossRefGoogle Scholar
Kyprianou, A. and Zhou, X. (2009). General tax structures and the Lévy insurance risk model. J. Appl. Prob. 46, 1146–1156.CrossRefGoogle Scholar
Landriault, D., Li, B. and Li, S. (2015). Analysis of a drawdown-based regime-switching Lévy insurance model. Insurance Math. Econom. 60, 98–107.10.1016/j.insmatheco.2014.11.005CrossRefGoogle Scholar
Landriault, D., Li, B. and Zhang, H. (2017). A unified approach for drawdown (drawup) of time-homogeneous Markov processes. J. Appl. Prob. 54, 603–626.10.1017/jpr.2017.20CrossRefGoogle Scholar
Lehoczky, J. (1977). Formulas for stopped diffusion processes with stopping times based on the maximum. Ann. Prob. 5, 601–607.10.1214/aop/1176995770CrossRefGoogle Scholar
Li, B., Tang, Q. and Zhou, X. (2013). A time-homogeneous diffusion model with tax. J. Appl. Prob. 50, 195–207.CrossRefGoogle Scholar
Li, B., Vu, L. and Zhou, X. (2019). Exit problems for general draw-down times of spectrally negative Lévy processes. J. Appl. Prob. 56, 441–457.CrossRefGoogle Scholar
Mijatović, A. and Pistorius, M. (2012). On the drawdown of completely asymmetric Lévy processes. Stoch. Process. Appl. 122, 3812–3836.10.1016/j.spa.2012.06.012CrossRefGoogle Scholar
Page, E. (1954). Continuous inspection schemes. Biometrika 41, 100–115.10.1093/biomet/41.1-2.100CrossRefGoogle Scholar
Shepp, L. and Shiryaev, A. (1993). The Russian option: reduced regret. Ann. Appl. Prob. 3, 631–640.10.1214/aoap/1177005355CrossRefGoogle Scholar
Taylor, H. (1975). A stopped Brownian motion formula. Ann. Prob. 3, 234–246.10.1214/aop/1176996395CrossRefGoogle Scholar
Zhang, H. (2018). Stochastic Drawdowns. World Scientific, Singapore.CrossRefGoogle Scholar
Zhou, J., Wu, L. and Bai, Y. (2017). Occupation times of Lévy-driven Ornstein–Uhlenbeck processes with two-sided exponential jumps and applications. Statist. Prob. Lett. 125, 80–90.CrossRefGoogle Scholar