Displaying similar documents to “Exponential functionals of brownian motion and class-one Whittaker functions”

On the exponential Orlicz norms of stopped Brownian motion

Goran Peškir (1996)

Studia Mathematica

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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...