Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-27T23:13:20.997Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundary crossing probabilities for high-dimensional Brownian motion

Part of: Markov processes

Published online by Cambridge University Press:  21 June 2016

James C. Fu  and
Tung-Lung Wu
James C. Fu*
Affiliation:
University of Manitoba
Tung-Lung Wu*
Affiliation:
Mississippi State University
*
* Postal address: Department of Statistics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada.
** Postal address: Department of Mathematics and Statistics, Mississippi State University, Starkville, MS 39759, USA. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The two-sided nonlinear boundary crossing probabilities for one-dimensional Brownian motion and related processes have been studied in Fu and Wu (2010) based on the finite Markov chain imbedding technique. It provides an efficient numerical method to computing the boundary crossing probabilities. In this paper we extend the above results for high-dimensional Brownian motion. In particular, we obtain the rate of convergence for high-dimensional boundary crossing probabilities. Numerical results are also provided to illustrate our results.

Keywords

Boundary crossing probabilityfinite Markov chain imbeddingBrownian motionY-type time tunnelirregular boundarytwo-dimensionalhigh-dimensionalrate of convergence

MSC classification

Primary: 60J65: Brownian motion
Secondary: 60J60: Diffusion processes 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 543 - 553
Copyright
Copyright © Applied Probability Trust 2016 

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

[1]Anderson, T. W. (1960).A modification of the sequential probability ratio test to reduce the sample size.Ann. Math. Statist. 31, 165–197.Google Scholar
[2]Buckholtz, P. G. and Wasan, M. T. (1979).First passage probabilities of a two-dimensional Brownian motion in an anisotropic medium.Sankhyā A 41, 198–206.Google Scholar
[3]Di Nardo, E., Nobile, A. G., Pirozzi, E. and Ricciardi, L. M. (2001).A computational approach to first-passage-time problems for Gauss–Markov processes.Adv. Appl. Prob. 33, 453–482.CrossRefGoogle Scholar
[4]Durbin, J. (1985).The first-passage density of a continuous Gaussian process to a general boundary.J. Appl. Prob. 22, 99–122. (Correction: 25 (1988), 804.) Google Scholar
[5]Durbin, J. (1992).The first-passage density of the Brownian motion process to a curved boundary.J. Appl. Prob. 29, 291–304.CrossRefGoogle Scholar
[6]Erdős, P. and Kac, M. (1946).On certain limit theorems of the theory of probability.Bull. Amer. Math. Soc. 52, 292–302.CrossRefGoogle Scholar
[7]Fu, J. C. and Wu, T.-L. (2010).Linear and nonlinear boundary crossing probabilities for Brownian motion and related processes.J. Appl. Prob. 47, 1058–1071.Google Scholar
[8]Iyengar, S. (1985).Hitting lines with two-dimensional Brownian motion.SIAM J. Appl. Math. 45, 983–989.Google Scholar
[9]Lehmann, A. (2002).Smoothness of first passage time distributions and a new integral equation for the first passage time density of continuous Markov processes.Adv. Appl. Prob. 34, 869–887.Google Scholar
[10]Metzler, A. (2010).On the first passage problem for correlated Brownian motion.Statist. Prob. Lett. 80, 277–284.Google Scholar
[11]Novikov, A., Frishling, V. and Kordzakhia, N. (1999).Approximations of boundary crossing probabilities for a Brownian motion.J. Appl. Prob. 36, 1019–1030.CrossRefGoogle Scholar
[12]Robbins, H. and Siegmund, D. (1970).Boundary crossing probabilities for the Wiener process and sample sums.Ann. Math. Statist. 41, 1410–1429.CrossRefGoogle Scholar
[13]Sacerdote, L. and Tomassetti, F. (1996).On evaluations and asymptotic approximations of first-passage-time probabilities.Adv. Appl. Prob. 28, 270–284.Google Scholar
[14]Scheike, T. H. (1992).A boundary-crossing result for Brownian motion.J. Appl. Prob. 29, 448–453.Google Scholar
[15]Spitzer, F. (1958).Some theorems concerning 2-dimensional Brownian motion.Trans. Amer. Math. Soc. 87, 187–197.Google Scholar
[16]Walsh, J. L. (1921).On the transformation of convex point sets.Ann. Math. (2) 22, 262–266.CrossRefGoogle Scholar
[17]Wendel, J. G. (1980).Hitting spheres with Brownian motion.Ann. Prob. 8, 164–169.Google Scholar