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• Shepp_DoubleBMV_newV12

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

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Journal publication date 31/03/2020 Extremes 1 23 20 35-54 Published 11/09/19 English

### Abstract

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.

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The final publication is available at Springer via http://dx.doi.org/10.1007/s10687-019-00357-z