# A lower bound for boundary crossing probabilities of Brownian bridge/motion with trend

### Details

Serval ID

serval:BIB_BE8DB3D5E2BD

Type

**Article**: article from journal or magazin.

Collection

Publications

Fund

Title

A lower bound for boundary crossing probabilities of Brownian bridge/motion with trend

Journal

Statistics & Probability Letters

ISSN

0167-7152

Publication state

Published

Issued date

2005

Peer-reviewed

Oui

Volume

74

Number

3

Pages

265-271

Language

english

Abstract

We show a lower bound for the boundary crossing probability P{there exists z is an element of [0, 1] : h(z) + B-0(z) > u(z)} with B-0 a Brownian bridge, h a trend function and u a boundary function. By that we get also a lower bound for the boundary crossing probability P{there exists z is an element of [0, 1] : h(z) + B-0(z) < u(z)}. It turns out that the bound improves the asymptotic result given in Bischoff et al. [2003a. Methodology Comput. Appl. Probab. 5 (3), 271-287] when considering a trend function gamma h, gamma -> infinity. The usual tools to obtain boundary crossing probabilities are finite dimensional approximations of the above probability. In this paper, however, we use the Cameron-Martin-Girsanov formula to obtain a bound of the above probability.

Keywords

Boundary crossing probability, Brownian bridge with trend, Brownian motion with trend, Signal-plus-noise model, Cameron-Martin-Girsanov formula

Web of science

Create date

03/09/2010 10:54

Last modification date

03/03/2018 20:00