Tail asymptotics for Shepp-statistics of Brownian motion in $\mathbb {R}^{d}$

Abstract

Let X(t), $t\in \mathbb {R}$, be a d-dimensional vector-valued Brownian motion, d ≥ 1. For all $\boldsymbol {b}\in \mathbb {R}^{d}\setminus (-\infty ,0]^{d}$ we derive exact asymptotics of

$$ \mathbb{P}\{\boldsymbol{X}(t+s)-\boldsymbol{X}(t) >u\boldsymbol{b}\text{ for some } t\in[0,T],\ s\in[0,1]\} \quad\text{as } u\to\infty, $$

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 → 0 or $T\to \infty$. Results for high level 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.

Appendix: Quadratic programming problems

The next result is known and formulated for instance in Dębicki et al. (2018).

Lemma A.1

Let Σ bea positive definite matrix of sized × dwithinverse Σ^− 1.If$\boldsymbol {b} \in \mathbb {R}^{d} \setminus (-\infty , 0]^{d}$,then the quadratic programming problem π_Σ(b) formulatedin Eq. 3 has a unique solution$\widetilde {\boldsymbol {b}}$andthere exists a unique non-empty index set$I\subseteq \{1{\ldots } d\}$withm ≤delementssuch that

$$ \begin{array}{@{}rcl@{}} \widetilde{\boldsymbol{b}}_{I} = \boldsymbol{b}_{I} , \text{ and if } J := \{ 1 {\ldots} d\}\setminus I \not = \emptyset, \text{ then } \widetilde{\boldsymbol{b}}_{J} &=& {\Sigma}_{JI} {\Sigma}_{II}^{-1} \boldsymbol{b}_{I} \geq \boldsymbol{b}_{J}, ~~{\Sigma}_{II}^{-1} \boldsymbol{b}_{I}>\boldsymbol{0}_I, \end{array} $$

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$$ \begin{array}{@{}rcl@{}} \min_{\boldsymbol{x} \geq \boldsymbol{b}}\boldsymbol{x}^{\top} {{\Sigma}^{-1}}\boldsymbol{x}= \widetilde{\boldsymbol{b}}^{\top} {{\Sigma}^{-1}} \widetilde{\boldsymbol{b}} &=& \boldsymbol{b}_{I}^{\top} {\Sigma}_{II}^{-1}\boldsymbol{b}_{I}>0. \end{array} $$

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Furthermore, for any$\boldsymbol {x}\in \mathbb {R}^{d}$we have

$$ \boldsymbol{x}^{\top} {{\Sigma}^{-1}} \widetilde{\boldsymbol{b}}= \boldsymbol{x}_I^{\top} {\Sigma}_{II}^{-1} \widetilde{\boldsymbol{b}}_I= \boldsymbol{x}_I^{\top} {\Sigma}_{II}^{-1}\boldsymbol{b}_I $$

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and if$\boldsymbol {b}= c \boldsymbol {1}, c\in (0,\infty )$, then 2 ≤|I|≤kandJis empty if Σ^− 1b > 0.