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On the exponential Orlicz norms of stopped Brownian motion

with some fixed constant . The method of proof relies upon Taylor expansion, Burkholder-Gundy's inequality, best constants in Doob's maximal inequality, Davis' best constants in the -inequalities for stopped Brownian motion, and estimates of the smallest and largest positive zero of Hermite polynomials. The results extend to cover the case of any continuous local martingale (by applying the time change method of Dubins and Schwarz).