Asymptotic estimates for the distribution of additive functionals of Brownian motion by the Wiener-Hopf factorization method
type: Item
instance of: scholarly article
title: Asymptotic estimates for the distribution of additive functionals of Brownian motion by the Wiener-Hopf factorization method
zbMATH Open document ID: 0876.60067
author: Yasuki Isozaki
published in: Journal of Mathematics of Kyoto University
publication date: 1997-11-18T00:00:00Z
review text: Let \((\tau_t,\xi_t)\) be a càdlàg process with stationary independent increments starting from \((0,0)\) with characteristic exponent \(\psi(\mu,\eta)\), The author proves a certain generalization of the Wiener-Hopf decomposition theorem for \(\psi(\mu,\eta)\), i.e. so called ``Spitzer-Rogozin identity in continuous time'', and uses it to investigate the asymptotics of the distribution of additive functionals of standard one-dimensional Brownian motion \(b(u)\). For instance, it is proved that \[ P\Biggl\{\sup_{0\leq t\leq A}\int_0^t \text{sgn}(b(u))|b(u)|^\alpha du\leq r\Biggr\} \asymp r^{1/2(\alpha+2)} A^{-1/4} \] as \(Ar^{-2/(\alpha+2)}\to \infty\). This result is compared with more precise results obtained by \textit{S. Watanabe} and \textit{Y. Isozaki} with the help of Kolmogorov diffusion and MeKean's formula in [Proc. Japan Acad., Ser. A 70, No. 9, 271-276 (1994; Zbl 0820.60066)].
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 950588
zbMATH Keywords: Brownian motion
zbMATH Keywords: additive functionals
zbMATH Keywords: factorization method
zbMATH Keywords: local time
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518613
OpenAlex ID: W1608735932
DOI: 10.1215/KJM/1250518613
description: scientific article; zbMATH DE number 950588