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On the exponential Orlicz norms of stopped Brownian motion

Goran Peškir — 1996

Studia Mathematica

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 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 m a x 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 m a x 0 t τ | B t | ψ 1 K E ( τ ) with some universal constant K > 0....

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