Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-27T21:15:56.303Z Has data issue: false hasContentIssue false
  • English
  • Français

First Passage Times of (Reflected) Ornstein-Uhlenbeck Processes Over Random Jump Boundaries

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Lijun Bo ,
Yongjin Wang  and
Xuewei Yang
Lijun Bo*
Affiliation:
Xidian University
Yongjin Wang*
Affiliation:
Nankai University
Xuewei Yang*
Affiliation:
Nankai University and University of Illinois
*
∗ Postal address: Department of Mathematics, Xidian University, Xi'an 710071, P. R. China.
∗∗ Postal address: School of Business, Nankai University, Tianjin 300071, P. R. China.
∗∗∗ Postal address: School of Mathematical Sciences, Nankai University, Tianjin 300071, P. R. China. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we study first passage times of (reflected) Ornstein-Uhlenbeck processes over compound Poisson-type boundaries. In fact, we extend the results of first rendezvous times of (reflected) Brownian motion and compound Poisson-type processes in Perry, Stadje and Zacks (2004) to the (reflected) Ornstein-Uhlenbeck case.

Keywords

First passage timeOrnstein-Uhlenbeck processreflecting barriercompound Poisson-type boundary

MSC classification

Secondary: 60J60: Diffusion processes 60K10: Applications (reliability, demand theory, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 723 - 732
Copyright
Copyright © Applied Probability Trust 2011 

References

Alili, L., Patie, P. and Pedersen, J. L. (2005). Representations of the first hitting time density of an Ornstein-Uhlenbeck process. Stoch. Models 21, 967–980.Google Scholar
Ata, B., Harrison, J. M. and Shepp, L. A. (2005). Drift rate control of a Brownian processing system. Ann. Appl. Prob. 15, 1145–1160.Google Scholar
Bo, L., Wang, Y. and Yang, X. (2010). First passage problems on reflected generalized Ornstein-Uhlenbeck processes and applications. Preprint.Google Scholar
Bo, L., Wang, Y. and Yang, X. (2011). Some integral functionals of reflected SDEs and their applications in finance. Quant. Finance 11, 343–348.Google Scholar
Bo, L., Zhang, L. and Wang, Y. (2006). On the first passage times of reflected OU processes with two-sided barriers. Queueing Systems 54, 313–316.Google Scholar
Bo, L., Tang, D., Wang, Y. and Yang, X. (2011). On the conditional default probability in a regulated market: a structural approach. Quant. Finance, 8 pp.Google Scholar
Borovkov, K. and Novikov, A. (2008). On exit times of Levy-driven Ornstein-Uhlenbeck processes. Statist. Prob. Lett. 78, 1517–1525.Google Scholar
Hadjiev, D. I. (1985). The first passage problem for generalized Ornstein-Uhlenbeck processes with non-positive Jumps. In Séminaire de Probabilités XIX (Lecture Notes Math. 1123), Springer, Berlin, pp. 80–90.Google Scholar
Harrison, J. M. (1985). {Brownian Motion and Stochastic Flow Systems}. John Wiley, New York.Google Scholar
Itô, K. and McKean, H. P. Jr. (1996). {Diffusion Processes and Their Sample Paths}. Springer, Berlin.Google Scholar
Karatzas, I. and Shreve, S. E. (1991). {Brownian Motion and Stochastic Calculus}. Springer, New York.Google Scholar
Li, Y., Wang, Y. and Yang, X. (2010). On the hitting time density for reflected OU processes: with an application to the regulated market. Preprint.Google Scholar
Linetsky, V. (2005). On the transition densities for reflected diffusions. Adv. Appl. Prob. 37, 435–460.Google Scholar
Loeffen, R. L. and Patie, P. (2010). Absolute ruin in the Ornstein-Uhlenbeck type risk model. Preprint. Available at http://arxiv.org.abs/1006.2712v1.Google Scholar
Patie, P. (2005). On a martingale associated to generalized Ornstein-Uhlenbeck processes and an application to finance. Stoch. Process. Appl. 115, 593–607.Google Scholar
Perry, D., Stadje, W. and Zacks, S. (2004). The first rendezvous time of Brownian motion and compound Poisson-type processes. J. Appl. Prob. 41, 1059–1070.Google Scholar
Protter, P. E. (2004). {Stochastic Integration and Differential Equations}. Springer,Google Scholar