On the exponential Orlicz norms of stopped Brownian motion

Peškir, Goran

Peškir, Goran

Studia Mathematica, Tome 119 (1996), p. 253-273 / Harvested from The Polish Digital Mathematics Library

Résumé

Necessary and sufficient conditions are found for the exponential Orlicz norm (generated by $ψ_{p} (x) = {e x p (| x |}^{p}) - 1$ with 0 < p ≤ 2) of $m a x_{0 \leq t \leq τ} | B_{t} |$ or $| B_{τ} |$ to be finite, where $B = {(B_{t})}_{t \geq 0}$ is a standard Brownian motion and τ is a stopping time for B. The conditions are in terms of the moments of the stopping time τ. For instance, we find that $∥ m a x_{0 \leq t \leq τ} | B_{t} | ∥_{ψ_{1}} < \infty$ as soon as $E (τ^{k}) = O (C^{k} k^{k})$ for some constant C > 0 as k → ∞ (or equivalently $∥ τ ∥_{ψ_{1}} < \infty$ ). In particular, if τ ∼ Exp(λ) or $| N (0, σ^{2}) |$ then the last condition is satisfied, and we obtain $∥ m a x_{0 \leq t \leq τ} | B_{t} | ∥_{ψ_{1}} \leq K \sqrt E (τ)$ with some universal constant K > 0. Moreover, this inequality remains valid for any class of stopping times τ for B satisfying $E (τ^{k}) \leq C {(E τ)}^{k} k^{k}$ for all k ≥ 1 with some fixed constant C > 0. The method of proof relies upon Taylor expansion, Burkholder-Gundy’s inequality, best constants in Doob’s maximal inequality, Davis’ best constants in the $L^{p}$ -inequalities for stopped Brownian motion, and estimates of the smallest and largest positive zero of Hermite polynomials. The results extend to the case of any continuous local martingale (by applying the time change method of Dubins and Schwarz).

Publié le : 1996-01-01

Zbl 0849.60039

EUDML-ID : urn:eudml:doc:216255

@article{bwmeta1.element.bwnjournal-article-smv117i3p253bwm,
     author = {Goran Pe\v skir},
     title = {On the exponential Orlicz norms of stopped Brownian motion},
     journal = {Studia Mathematica},
     volume = {119},
     year = {1996},
     pages = {253-273},
     zbl = {0849.60039},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv117i3p253bwm}
}

Peškir, Goran. On the exponential Orlicz norms of stopped Brownian motion. Studia Mathematica, Tome 119 (1996) pp. 253-273. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv117i3p253bwm/

Bibliographie

[00000] [1] M. Abramowitz and I. A. Stegun, Pocketbook of Mathematical Functions (abridged edition of Handbook of Mathematical Functions, National Bureau of Standards, 1964), Verlag Harri Deutsch, Thun-Frankfurt/Main, 1984.

[00001] [2] D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249-304. | Zbl 0223.60021

[00002] [3] B. Davis, On the $L^{p}$ norms of stochastic integrals and other martingales, Duke Math. J. 43 (1976), 697-704. | Zbl 0349.60061

[00003] [4] L. E. Dubins and G. Schwarz, On continuous martingales, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 913-916. | Zbl 0203.17504

[00004] [5] A. Gut, Stopped Random Walks (Limit Theorems and Applications), Springer, 1988.

[00005] [6] U. Haagerup, The best constants in the Khintchine inequality, Studia Math. 70 (1982), 231-283. | Zbl 0501.46015

[00006] [7] M. Ledoux and M. Talagrand, Probability in Banach Spaces (Isoperimetry and Processes), Springer, Berlin, 1991. | Zbl 0748.60004

[00007] [8] R. Sh. Liptser and A. N. Shiryayev, Theory of Martingales, Kluwer Academic Publ., 1989. | Zbl 0728.60048

[00008] [9] H. P. McKean, Stochastic Integrals, Wiley, 1969.

[00009] [10] A. A. Novikov, On stopping times for a Wiener process, Theory Probab. Appl. 16 (1971), 449-456. | Zbl 0258.60037

[00010] [11] G. Peškir, Best constants in Kahane-Khintchine inequalities in Orlicz spaces, J. Multivariate Anal. 45 (1993), 183-216. | Zbl 0773.41026

[00011] [12] G. Peškir, Maximal inequalities of Kahane-Khintchine's type in Orlicz spaces, Math. Proc. Cambridge Philos. Soc. 115 (1994), 175-190. | Zbl 0809.60005

[00012] [13] G. Peškir, Best constants in Kahane-Khintchine inequalities for complex Steinhaus variables, Proc. Amer. Math. Soc., to appear. | Zbl 0841.41024

[00013] [14] L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales, Vol. 2, Ito's Calculus, Wiley, 1987. | Zbl 0627.60001

[00014] [15] L. A. Shepp, A first passage problem for the Wiener process, Ann. Math. Statist. 38 (1967), 1912-1914. | Zbl 0178.19402

[00015] [16] G. Wang, Sharp maximal inequalities for conditionally symmetric martingales and Brownian motion, Proc. Amer. Math. Soc. 112 (1991), 579-586. | Zbl 0719.60050