Advances in Applied Probability

Estimates of the exit probability for two correlated Brownian motions

Jinghai Shao and Xiuping Wang

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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 ≤st Xsa, max 0 ≤st Ysb) 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 (Xtat+c,Ytbt+d, 0≤ tT) for any constants a, b≥0 and c,d, T >0.

Article information

Adv. in Appl. Probab., Volume 45, Number 1 (2013), 37-50.

First available in Project Euclid: 15 March 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

boundary crossing probability correlated Brownian motion exit probability reflection principle


Shao, Jinghai; Wang, Xiuping. Estimates of the exit probability for two correlated Brownian motions. Adv. in Appl. Probab. 45 (2013), no. 1, 37--50. doi:10.1239/aap/1363354102.

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