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Boundary crossing probabilities for high-dimensional Brownian motion

  • James C. Fu (a1) and Tung-Lung Wu (a2)

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.

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Corresponding author

* 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: comehome1981@gmail.com

References

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[1]Anderson, T. W. (1960).A modification of the sequential probability ratio test to reduce the sample size.Ann. Math. Statist. 31, 165197.
[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, 198206.
[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, 453482.
[4]Durbin, J. (1985).The first-passage density of a continuous Gaussian process to a general boundary.J. Appl. Prob. 22, 99122. (Correction: 25 (1988), 804.)
[5]Durbin, J. (1992).The first-passage density of the Brownian motion process to a curved boundary.J. Appl. Prob. 29, 291304.
[6]Erdős, P. and Kac, M. (1946).On certain limit theorems of the theory of probability.Bull. Amer. Math. Soc. 52, 292302.
[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, 10581071.
[8]Iyengar, S. (1985).Hitting lines with two-dimensional Brownian motion.SIAM J. Appl. Math. 45, 983989.
[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, 869887.
[10]Metzler, A. (2010).On the first passage problem for correlated Brownian motion.Statist. Prob. Lett. 80, 277284.
[11]Novikov, A., Frishling, V. and Kordzakhia, N. (1999).Approximations of boundary crossing probabilities for a Brownian motion.J. Appl. Prob. 36, 10191030.
[12]Robbins, H. and Siegmund, D. (1970).Boundary crossing probabilities for the Wiener process and sample sums.Ann. Math. Statist. 41, 14101429.
[13]Sacerdote, L. and Tomassetti, F. (1996).On evaluations and asymptotic approximations of first-passage-time probabilities.Adv. Appl. Prob. 28, 270284.
[14]Scheike, T. H. (1992).A boundary-crossing result for Brownian motion.J. Appl. Prob. 29, 448453.
[15]Spitzer, F. (1958).Some theorems concerning 2-dimensional Brownian motion.Trans. Amer. Math. Soc. 87, 187197.
[16]Walsh, J. L. (1921).On the transformation of convex point sets.Ann. Math. (2) 22, 262266.
[17]Wendel, J. G. (1980).Hitting spheres with Brownian motion.Ann. Prob. 8, 164169.

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Boundary crossing probabilities for high-dimensional Brownian motion

  • James C. Fu (a1) and Tung-Lung Wu (a2)

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