Small noise asymptotics and first passage times of integrated Ornstein-Uhlenbeck processes driven by α-stable Lévy processes

Bernoulli

Volumn 20, Issue 1, 2014, Pages 265-281

  Hintze, Robert ^a   Pavlyukevich, Ilya ^a  

^a FRIEDRICH SCHILLER UNIVERSITY JENA   (Germany)

Author keywords

stable L vy process; Absolutely continuous approximations; First passage times; Integrated Ornstein Uhlenbeck process; Skorokhod M1 topology; Tightness

Indexed keywords

EID: 84893678968     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/12-BEJ485     Document Type: Article

References (17)

1. Adler, R.J. (1990). An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. Institute of Mathematical Statistics Lecture Notes-Monograph Series 12. Hayward, CA: IMS. MR1088478
2. Al-Talibi, H., Hilbert, A. and Kolokoltsov, V. (2010). Nelson-type limit for a particular class of Lévy processes. AIP Conf. Proc. 1232 189.
3. Applebaum, D. (2009). Extending stochastic resonance for neuron models to general Lèvy noise. IEEE Transactions on Neural Networks 20 1993-1995.
4. Barndorff-Nielsen, O.E. (1998). Processes of normal inverse Gaussian type. Finance Stoch. 2 41-68. MR1804664
5. Barndorff-Nielsen, O.E. and Shephard, N. (2003). Integrated OU processes and non-Gaussian OUbased stochastic volatility models. Scand. J. Stat. 30 277-295. MR1983126
6. Chechkin, A.V., Gonchar, V.Y. and Szydlowski, M. (2002). Fractional kinetics for relaxation and superdiffusion in a magnetic field. Physics of Plasmas 9 78-88.
7. Garbaczewski, P. and Olkiewicz, R. (2000). Ornstein-Uhlenbeck-Cauchy process. J. Math. Phys. 41 6843-6860. MR1781410
8. Hesse, C.H. (1991). The one-sided barrier problem for an integrated Ornstein-Uhlenbeck process. Comm. Statist. Stochastic Models 7 447-480. MR1123138
9. Horsthemke, W. and Lefever, R. (1984). Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology. Springer Series in Synergetics 15. Berlin: Springer. MR0724433
10. Kuznetsov, A. (2011). On extrema of stable processes. Ann. Probab. 39 1027-1060. MR2789582
11. Lefebvre, M. (1989). Moment generating function of a first hitting place for the integrated Ornstein- Uhlenbeck process. Stochastic Process. Appl. 32 281-287. MR1014454
12. Puhalskii, A.A. and Whitt, W. (1997). Functional large deviation principles for first-passage-time processes. Ann. Appl. Probab. 7 362-381. MR1442318
13. Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge: Cambridge Univ. Press. Translated from the 1990 Japanese original, revised by the author. MR1739520
14. Simon, T. (2011). Hitting densities for spectrally positive stable processes. Stochastics 83 203-214. MR2800088
15. Skorohod, A.V. (1956). Limit theorems for stochastic processes. Theory Probab. Appl. 1 261-290. MR0084897
16. Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and their Application to Queues. Springer Series in Operations Research. New York: Springer. MR1876437
17. Williams, D.R.E. (2001). Path-wise solutions of stochastic differential equations driven by Lévy processes. Rev. Mat. Iberoam. 17 295-329. MR1891200