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≤t Xs≥ a, max 0 ≤s≤t Ys≥ 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.
"Estimates of the exit probability for two correlated Brownian motions." Adv. in Appl. Probab. 45 (1) 37 - 50, March 2013. https://doi.org/10.1239/aap/1363354102