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 be the Ornstein-Uhlenbeck velocity process solving

with , where and is a standard Brownian motion. Then there exist universal constants and such that

for all stopping times of . In particular, this yields the existence of universal constants and such that

for all stopping times of . 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.

**[1]**D. L. Burkholder and R. F. Gundy,*Extrapolation and interpolation of quasi-linear operators on martingales*, Acta Math.**124**(1970), 249–304. MR**0440695****[2]**E. Lenglart,*Relation de domination entre deux processus*, Ann. Inst. H. Poincaré Sect. B (N.S.)**13**(1977), no. 2, 171–179 (French, with English summary). MR**0471069****[3]**Edward Nelson,*Dynamical theories of Brownian motion*, Princeton University Press, Princeton, N.J., 1967. MR**0214150****[4]**Peskir, G. (1998). Controlling the velocity of Brownian motion by its terminal value.*Research Report*No. 391,*Dept. Theoret. Statist. Aarhus*(11 pp).*Analytic and Geometric Inequalities and their Applications*(eds. T. M. Rassias and H. M. Srivastava), Math. Appl., Vol. 478, Kluwer Acad. Publ., Dordrecht, 1999 (323-333).**[5]**Daniel Revuz and Marc Yor,*Continuous martingales and Brownian motion*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1994. MR**1303781****[6]**Uhlenbeck, G. E. and Ornstein, L. S. (1930). On the theory of Brownian motion.*Physical Review*36 (823-841).

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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

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