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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Maximal inequalities for the Ornstein-Uhlenbeck process
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by S. E. Graversen and G. Peskir PDF
Proc. Amer. Math. Soc. 128 (2000), 3035-3041 Request permission

Abstract:

Let $V=(V_t)_{t \ge 0}$ be the Ornstein-Uhlenbeck velocity process solving \[ dV_t = - \beta V_t dt + dB_t\] 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 \[ \frac {C_1}{\sqrt { \beta }} E\sqrt { \log (1+ \beta \tau )} \le E\bigg (\max _{0 \le t \le \tau } \vert V_t \vert \bigg ) \le \frac {C_2}{\sqrt {\beta }} E\sqrt { \log (1+ \beta \tau )}\] 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 \[ D_1 E\sqrt { \log \big (1+\log (1+ \tau )\big )} \le E\bigg (\max _{0 \le t \le \tau } \frac {\vert B_t \vert }{\sqrt {1+t }} \bigg ) \le D_2 E\sqrt { \log \big (1+\log (1+ \tau )\big )}\] 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
  • D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249–304. MR 440695, DOI 10.1007/BF02394573
  • 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
  • Edward Nelson, Dynamical theories of Brownian motion, Princeton University Press, Princeton, N.J., 1967. MR 0214150
  • 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).
  • 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
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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)
  • MR Author ID: 337521
  • Email: goran@imf.au.dk
  • 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
  • © Copyright 2000 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 128 (2000), 3035-3041
  • MSC (2000): Primary 60J65, 60G40, 60E15; Secondary 60J60, 60G15
  • DOI: https://doi.org/10.1090/S0002-9939-00-05345-4
  • MathSciNet review: 1664394