On the Joint Extremes of Two Correlated Fractional Brownian Motions

Let X_i,i=1,2 be two standard fractional Brownian motions with constant joint correlation. In this paper we derive the exact asymptotics of the joint survival function Prob{sup_{s\in[0,1]} X_1(s)>u, sup_{t\in[0,1]}X_2(t)>u} as u tends to infinity. As a by-product we obtain generalizations of the Borell-TIS inequality and the Piterbarg inequality for 2-dimensional Gaussian random fields.