On an approach to boundary crossing by stochastic processes

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:126

Article

Brown, Mark¹  de la Pena, Victor H.¹  Klass, Michael J.²  Sit, Tony³ 

[1] Columbia Univ, Dept Stat, New York, NY 10027 USA

[2] Univ Calif Berkeley, Dept Math & Stat, Berkeley, CA 94720 USA

[3] Chinese Univ Hong Kong, Dept Stat, Hong Kong, Hong Kong, Peoples R China

关键词: First-hitting time;    Renewal theory;    Decoupling;    Probability bounds;   

DOI  :  10.1016/j.spa.2016.04.027

来源: Elsevier

【 摘 要 】

In this paper we provide an overview as well as new (definitive) results of an approach to boundary crossing. The first published results in this direction appeared in de la Pena and Gine (1999) book on decoupling. They include order of magnitude bounds for the first hitting time of the norm of continuous Banach-Space valued processes with independent increments. One of our main results is a sharp lower bound for the first hitting time of cadlag real-valued processes X (t), where X (0) = 0 with arbitrary dependence structure: ETr gamma >= integral(1)(0) {a(-1)(r alpha)}(gamma) d alpha, where T-r = inf{t > 0 : X (t) >= r}, a(t) = E{sup(0 <= s <= t) X (s)} and gamma > 0. Under certain extra conditions, we also obtain an upper bound for ETr gamma. As the main text suggests, although T-r is defined as the hitting time of X (t) hitting a level boundary, the bounds developed can be extended to more general processes and boundaries. We shall illustrate applications of the bounds derived for additive processes, Gaussian Processes, Bessel Processes, Bessel bridges among others. By considering the non-random function a(t), we can show that in various situations, ETr approximate to a(-1)(r). (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

Free