Abstract
Markov-modulated Lévy processes with two different regimes of restarting are studied. These regimes correspond to the completely renewed process and to the process of Markov modulation, accompanied by jumps. We give explicit expressions for the Lévy–Khintchine exponent in the case of a two-state underlying Markov chain. For the renewal case, the limit distributions (as \(t\rightarrow \infty \)) are obtained. In the case of processes with jumps, we present some results for the exponential functional.
Similar content being viewed by others
References
Asmussen, S.: Applied probability and queues. In: Applications of Mathematics: Stochastic modeling and Applied Probability. Springer. (2003). https://doi.org/10.1080/15326349.2014.958424
Bertoin, J., Biane, Ph., Yor, M.: Poissonian exponential functionals, \(q\)-series, \(q\)-integrals, and the moment problem for log-normal distributions. In: Progress in Probability, vol. 58, pp. 45–56 (2004) https://link.springer.com/chapter/10.1007/978-3-0348-7943-9-3
Bertoin, J., Yor, M.: Exponential functionals of Lévy processes. Probab. Surv. 2, 191–212 (2005). https://doi.org/10.1214/154957805100000122
Bogachev, L., Ratanov, N.: Occupation time distributions for the telegraph processes. Stoch. Process. Appl. 121, 1816–1844 (2011). https://doi.org/10.1016/j.spa.2011.03.016
Boxma, O., Perry, D., Stadje, W., Zacks, S.: A Markovian growth-collapse model. Adv. Appl. Probab. 38, 221–243 (2006). https://doi.org/10.1239/aap/1143936148
Boyarchenko, S.I., Levendorskiǐ, S.Z.: American options in regime-switching models. SIAM J. Control Optim. 48(3), 1353–1376 (2009). https://doi.org/10.1137/070682897
Carmona, Ph., Petit, F., Yor, M.: Exponential functionals of Lévy processes. In: Barndorff-Nielsen, O.E., Mikosh, T., Resnick, S.I (eds.) Lévy processes. Theory and Applications, pp. 41–55. Springer, Berlin (2001)
Cai, N., Kou, S.: Pricing asian options under a hyper-exponential jump diffusion model. Oper. Res. 60(1), 64–77 (2012). https://doi.org/10.1287/opre.1110.1006
Chevallier, J., Goutte, S.: On the estimation of regime-switching Lévy models. Stud. Nonlinear Dyn. E. 21(1), 3–29 (2017). https://doi.org/10.1515/snde-2016-0048
Chourdakis, K.: Switching Lévy models in continuous time: finite distributions and option pricing. SSRN Electronic Journal. University of Essex, Centre for Computational Finance and Economic Agents (CCFEA) Working Paper (2005). https://ssrn.com/abstract=838924 https://doi.org/10.2139/ssrn.838924
Di Nunno, G., Øksendal, B., Proske, F.: Malliavin Calculus for Lévy Processes with Applications to Finance. Springer, Berlin (2009). https://doi.org/10.1007/978-3-540-78572-9
Elliott, R.J., Chan, L., Siu, T.K.: Option pricing and Esscher transform under regime switching. Ann Finance 1, 423–432 (2005). https://doi.org/10.1007/s10436-005-0013-z
Epps, Th W.: Pricing Derivative Securities. World Scientific, Singapore (2007). https://doi.org/10.1142/9789812792914
Goldstein, S.: On diffusion by discontinuous movements and on the telegraph equation. Q. J. Mech. Appl. Math. 4, 129–156 (1951). https://doi.org/10.1093/qjmam/4.2.129
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products. Academic Press, Boston (1994)
Hainaut, D.: Financial modeling with switching Lévy processes. ESC Rennes Business School & CREST, France (2011) www.crest.fr/ckfinder/.../files/.../HainautSwitchingLevyProcess.pdf
Hainaut, D., Le Courtois, O.: An intensity model for credit risk with switching Lévy processes. Quant. Finance 14(8), 1453–1465 (2014). https://doi.org/10.1080/14697688.2012.756583
Kac, M.: A stochastic model related to the telegrapher’s equation. Rocky Mt. J Math. 4, 497–509 (1974). https://doi.org/10.1216/RMJ-1974-4-3-497. Reprinted from: M. Kac, Some stochastic problems in physics and mathematics. Colloquium lectures in the pure and applied sciences, No. 2, hectographed, Field Research Laboratory, Socony Mobil Oil Company, Dallas, TX, 1956, pp. 102-122
Kallenberg, O.: Foundations of Modern Probability. Springer (1997). www.springer.com/us/book/9780387953137
Kolesnik, A.D., Ratanov, N.: Telegraph Processes and Option Pricing. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40526-6
Kuznetsov, A.: On the distribution of exponential functionals for Lévy processes with jumps of rational transform. Stoch. Proc. Appl. 122, 654–663 (2012). https://doi.org/10.1016/j.spa.2011.09.007
Kuznetsov, A., Kyprianou, A.E., Pardo, J.C., Watson, A.R.: The hitting time of zero for a stable process. Electron. J. Probab. 19(30), 1–26 (2014). https://doi.org/10.1214/EJP.v19-2647
Kyprianou, A.E.: Fluctuations of Lévy Processes with Applications, Introductory Lectures. Universitext, Second edn. Springer, Berlin (2014). https://doi.org/10.1007/978-3-642-37632-0_2
Prudnikov, A.P., Brychkov, Yu, A., Marichev, O.I.: Integrals and series, vol. 4. Direct Laplace Transforms. Gordon and Breach Science Pub (1992)
Ratanov, N.: Piecewise linear process with renewal starting points. Stat. Probab. Lett. 131, 78–86 (2017). https://doi.org/10.1016/j.spl.2017.08.010
Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.: Stoch. Proces. Insur. Finance. Wiley, Hoboken (1999). https://doi.org/10.1002/9780470317044
López, O., Ratanov, N.: Kac’s rescaling for jump-telegraph processes. Stat. Probab. Lett. 82, 1768–1776 (2012). https://doi.org/10.1016/j.spl.2012.05.024
López, O., Ratanov, N.: On the asymmetric telegraph processes. J. Appl. Prob. 51(2), 569–589 (2014). https://doi.org/10.1239/jap/1402578644
Royal, A.J., Elliott, R.J.: Asset prices with regime-switching variance gamma dynamics. In: P.G. Ciarlet (Ed.) Mathematical Modeling and Numerical Methods in Finance. Special Volume (Alain Bensoussan and Qiang Zhang, Guest Editors) of Handbook of Numerical Analysis, VOL. XV ISSN 1570-8659, pp. 685–711. (2009). https://doi.org/10.1016/S1570-8659(08)00018-5
Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)
Shanthikumar, J., Sumita, U.: General shock models associated with correlated renewal sequences. J. Appl. Probab. 20, 600–614 (1983). https://doi.org/10.1017/S0021900200023858
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ratanov, N. Kac–Lévy Processes. J Theor Probab 33, 239–267 (2020). https://doi.org/10.1007/s10959-018-0873-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-018-0873-6
Keywords
- Markov-modulated Lévy process
- Markov-switching model
- Goldstein–Kac process
- Lévy–Khintchine exponent
- Lévy–Laplace exponent
- Mixture of distributions
- Exponential functional