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Small and large time stability of the time taken for a Lévy process to cross curved boundaries

Philip S. GriffinRoss A. Maller — 2013

Annales de l'I.H.P. Probabilités et statistiques

This paper is concerned with the small time behaviour of a Lévy process X . In particular, we investigate theof the times, T ¯ b ( r ) and T b * ( r ) , at which X , started with X 0 = 0 , first leaves the space-time regions { ( t , y ) 2 : y r t b , t 0 } (one-sided exit), or { ( t , y ) 2 : | y | r t b , t 0 } (two-sided exit), 0 b l t ; 1 , as r 0 . Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in L p . In many instances these are seen to be...

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