## Maximal inequalities for the Ornstein-Uhlenbeck process

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- by S. E. Graversen and G. Peskir
- Proc. Amer. Math. Soc.
**128**(2000), 3035-3041 - DOI: https://doi.org/10.1090/S0002-9939-00-05345-4
- Published electronically: April 7, 2000
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## 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**

## Bibliographic 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