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1995-1996 | 117 | 3 | 253-273
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On the exponential Orlicz norms of stopped Brownian motion

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Necessary and sufficient conditions are found for the exponential Orlicz norm (generated by $ψ_p(x) = exp(|x|^p)-1$ with 0 < p ≤ 2) of $max_{0≤t≤τ}|B_t|$ or $|B_τ|$ to be finite, where $B = (B_t)_{t≥0}$ 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 $∥max_{0≤t≤τ}|B_t|∥_{ψ_1} < ∞$ as soon as $E(τ^{k}) = O(C^{k}k^{k})$ for some constant C > 0 as k → ∞ (or equivalently $∥τ∥_{ψ_1} < ∞$). In particular, if τ ∼ Exp(λ) or $|N(0,σ^2)|$ then the last condition is satisfied, and we obtain $∥max_{0≤t≤τ}|B_t|∥_{ψ_1} ≤ K √{E(τ)}$ with some universal constant K > 0. Moreover, this inequality remains valid for any class of stopping times τ for B satisfying $E(τ^{k}) ≤ C(Eτ)^{k}k^{k}$ 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).
Twórcy
  • Institute of Mathematics, University of Aarhus, Ny Munkegade, 8000 Aarhus, Denmark , goran@mi.aau.dk
  • Department of Mathematics, University of Zagreb, Bijenička 30, 41000 Zagreb, Croatia
Bibliografia
  • [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.
  • [2] D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249-304.
  • [3] B. Davis, On the $L^p$ norms of stochastic integrals and other martingales, Duke Math. J. 43 (1976), 697-704.
  • [4] L. E. Dubins and G. Schwarz, On continuous martingales, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 913-916.
  • [5] A. Gut, Stopped Random Walks (Limit Theorems and Applications), Springer, 1988.
  • [6] U. Haagerup, The best constants in the Khintchine inequality, Studia Math. 70 (1982), 231-283.
  • [7] M. Ledoux and M. Talagrand, Probability in Banach Spaces (Isoperimetry and Processes), Springer, Berlin, 1991.
  • [8] R. Sh. Liptser and A. N. Shiryayev, Theory of Martingales, Kluwer Academic Publ., 1989.
  • [9] H. P. McKean, Stochastic Integrals, Wiley, 1969.
  • [10] A. A. Novikov, On stopping times for a Wiener process, Theory Probab. Appl. 16 (1971), 449-456.
  • [11] G. Peškir, Best constants in Kahane-Khintchine inequalities in Orlicz spaces, J. Multivariate Anal. 45 (1993), 183-216.
  • [12] G. Peškir, Maximal inequalities of Kahane-Khintchine's type in Orlicz spaces, Math. Proc. Cambridge Philos. Soc. 115 (1994), 175-190.
  • [13] G. Peškir, Best constants in Kahane-Khintchine inequalities for complex Steinhaus variables, Proc. Amer. Math. Soc., to appear.
  • [14] L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales, Vol. 2, Ito's Calculus, Wiley, 1987.
  • [15] L. A. Shepp, A first passage problem for the Wiener process, Ann. Math. Statist. 38 (1967), 1912-1914.
  • [16] G. Wang, Sharp maximal inequalities for conditionally symmetric martingales and Brownian motion, Proc. Amer. Math. Soc. 112 (1991), 579-586.
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Bibliografia
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