Running supremum of Brownian motion in dimension 2: exact and asymptotic results

This note investigates πT,c1,c2(a1,a2)=P(supt∈[0,T](σ1B(t)−c1t)>a1,supt∈[0,T](σ2B(t)−c2t)>a2), where {B(t):t≥0} is a standard Brownian motion, with T>0,σ1,σ2>0,c1,c2∈R. We derive explicit formula for the probability πT,c1,c2(a1,a2) and find its asymptotic behavior both in so called many-source high-threshold regimes.