Tail asymptotics for Shepp-statistics of Brownian motion in ℝ d

Let $X(t)$, $t\in R$, be a $d$-dimensional vector-valued Brownian motion, $d\ge 1$. For all $b\in R^d\setminus (-\infty,0]^d$ we derive exact asymptotics of \[ P(X(t+s)-X(t) >u b\mbox{ for some }t\in[0,T],\ s\in[0,1]}\mbox{as }u\to\infty, \] that is the asymptotical behavior of tail distribution of vector-valued analog of Shepp-statistics for $X$; we cover not only the case of a fixed time-horizon $T>0$ but also cases where $T\to 0$ or $T\to\infty$. Results for high excursion probabilities of vector-valued processes are rare in the literature, with currently no available approach suitable for our problem. Our proof exploits some distributional properties of vector-valued Brownian motion, and results from quadratic programming problems. As a by-product we derive a new inequality for the `supremum' of vector-valued Brownian motions