Extremes and first passage times of correlated fBm's

Abstract

Let $\{X_i(t),t\ge 0\},i=1,2$ be two standard fractional Brownian motions being jointly Gaussian with constant cross-correlation. In this paper we derive the exact asymptotics of the joint survival function $$\mathbb{P}\{\underset{s\in [0,1]}{\sup }{X}_1(s)>u,\underset{t\in [0,1]}{\sup }{X}_2(t)>u\}$$ as $u\to \infty$. A novel finding of this contribution is the exponential approximation of the joint conditional first passage times of $X_1,X_2$. As a by-product we obtain generalizations of the Borell-TIS inequality and the Piterbarg inequality for 2-dimensional Gaussian random fields.

16 pages, title changed

References in corpus (1)

• Asymptotics for Kotz Type III Elliptical Distributions