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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Maximal inequalities for the Ornstein-Uhlenbeck process


Authors: S. E. Graversen and G. Peskir
Journal: Proc. Amer. Math. Soc. 128 (2000), 3035-3041
MSC (2000): Primary 60J65, 60G40, 60E15; Secondary 60J60, 60G15
Published electronically: April 7, 2000
MathSciNet review: 1664394
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $V=(V_t)_{t \ge 0}$be the Ornstein-Uhlenbeck velocity process solving

\begin{displaymath}dV_t = - \beta V_t dt + dB_t\end{displaymath}

with $V_0=0$ , where $ \beta >0$ and $B=(B_t)_{t \ge 0}$ is a standard Brownian motion. Then there exist universal constants $C_1>0$and $C_2>0$ such that

\begin{displaymath}\frac{C_1}{\sqrt{ \beta }} E\sqrt{ \log (1+ \beta \tau )} \le... ...) \le \frac{C_2}{\sqrt{\beta }} E\sqrt{ \log (1+ \beta \tau )}\end{displaymath}

for all stopping times $\tau $ of $V$ . In particular, this yields the existence of universal constants $D_1>0$ and $D_2>0$ such that

\begin{displaymath}D_1 E\sqrt{ \log\big(1+\log (1+ \tau )\big)} \le E\bigg(\max_... ...{1+t }} \bigg) \le D_2 E\sqrt{ \log\big(1+\log (1+ \tau )\big)}\end{displaymath}

for all stopping times $\tau $ of $B$. This inequality may be viewed as a stopped law of iterated logarithm. The method of proof relies upon a variant of Lenglart's domination principle and makes use of Itô calculus.


References [Enhancements On Off] (What's this?)

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Additional Information

S. E. Graversen
Affiliation: Institute of Mathematics, University of Aarhus, Ny Munkegade, 8000 Aarhus, Denmark
Email: matseg@imf.au.dk

G. Peskir
Affiliation: Institute of Mathematics, University of Aarhus, Ny Munkegade, 8000 Aarhus, Denmark (Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia)
Email: goran@imf.au.dk

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05345-4
PII: S 0002-9939(00)05345-4
Keywords: Ornstein-Uhlenbeck velocity process, maximum process, stopping time, maximal inequality, Lenglart's domination principle, Brownian motion, diffusion process, Gaussian process, the Langevin stochastic differential equation
Received by editor(s): May 29, 1998
Received by editor(s) in revised form: November 10, 1998
Published electronically: April 7, 2000
Additional Notes: The authors were supported by the Danish National Research Foundation
Communicated by: Stanley Sawyer
Article copyright: © Copyright 2000 American Mathematical Society