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Estimates of the Exit Probability for Two Correlated Brownian Motions

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  04 January 2016

Jinghai Shao  and
Xiuping Wang
Jinghai Shao*
Affiliation:
Beijing Normal University
Xiuping Wang*
Affiliation:
Beijing Normal University
*
∗ Postal address: School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China. Email address: [email protected]
∗∗ Postal address: Beijing Normal University, Xin Jie Kou Wai Da Jie 19, Beijing 100875, China. Email address: [email protected]
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Abstract

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Given two correlated Brownian motions (Xt)t≥ 0 and (Yt)t≥ 0 with constant correlation coefficient, we give the upper and lower estimations of the probability ℙ(max0 ≤s≤tXs≥ a, max0 ≤s≤tYs≥ b) for any a,b,t > 0 through explicit formulae. Our strategy is to establish a new reflection principle for two correlated Brownian motions, which can be viewed as an extension of the reflection principle for one-dimensional Brownian motion. Moreover, we also consider the nonexit probability for linear boundaries, i.e. ℙ (Xt ≤ at+c,Yt ≤ bt+d, 0≤ t≤T) for any constants a, b≥0 and c,d, T > 0.

Keywords

boundary crossing probabilitycorrelated Brownian motionexit probabilityreflection principle

MSC classification

Primary: 60J65: Brownian motion
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 1 , March 2013 , pp. 37 - 50
Copyright
© Applied Probability Trust 

References

Anderson, T. W. (1960). A modification of the sequential probability ratio test to reduce the sample size. Ann. Math. Statist. 31, 165–197.CrossRefGoogle Scholar
Doob, J. L. (1949). Heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Statist. 20, 393–403.CrossRefGoogle Scholar
Durbin, J. (1992). The first-passage density of the Brownian motion process to a curved boundary. J. Appl. Prob. 29, 291–304.CrossRefGoogle Scholar
Feller, W. (1966). An introduction to Probability Theory and Its Applications, Vol. 2. John Wiley, New York.Google Scholar
Friedman, A. (2006). Stochastic Differential Equations and Applications. Dover Publications, Mineola, NY.Google Scholar
Iyengar, S. (1985). Hitting lines with two-dimensional Brownian motion. SIAM J. Appl. Math. 45, 983–989.CrossRefGoogle Scholar
Metzler, A. (2010). On the first passage problem for correlated Brownian motion. Statist. Prob. Lett. 80, 277–284.CrossRefGoogle Scholar
Novikov, A. A. (1981). On estimates and the asymptotic behavior of nonexit probabilities of a Wiener process to a moving boundary. Math. USSR SB 38, 495–505.CrossRefGoogle Scholar
Ricciardi, L. M., Sacerdote, L. and Sato, S. (1984). On an integral equation for first-passage-time probability densities. J. Appl. Prob. 21, 302–314.CrossRefGoogle Scholar
Siegmund, D. (1986). Boundary crossing probabilities and statistical applications. Ann. Math. Statist. 14, 361–404.Google Scholar
Wang, L. and Pötzelberger, K. (1997). Boundary crossing probability for Brownian motion and general boundaries. J. Appl. Prob. 34, 54–65.CrossRefGoogle Scholar
Pötzelberger, K. and Wang, L. (2001). Boundary crossing probability for Brownian motion. J. Appl. Prob. 38, 152–164.CrossRefGoogle Scholar