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Estimates of the Exit Probability for Two Correlated Brownian Motions

Part of: Stochastic processes Markov 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: shaojh@bnu.edu.cn
∗∗ Postal address: Beijing Normal University, Xin Jie Kou Wai Da Jie 19, Beijing 100875, China. Email address: wangxp@yahoo.com
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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 

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