Estimates of the exit probability for two correlated Brownian motions

Abstract

Given two correlated Brownian motions (X_t)_{t≥ 0} and (Y_t)_{t≥ 0} with constant correlation coefficient, we give the upper and lower estimations of the probability ℙ(max_{0 ≤s≤t} X_s≥ a, max _{0 ≤s≤t} Y_s≥ 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 (X_t ≤ at+c,Y_t ≤ bt+d, 0≤ t≤T) for any constants a, b≥0 and c,d, T >0.