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Stability of the Exit Time for L\'evy Processes

Abstract

This paper is concerned with the behaviour of a L\'{e}vy process when it crosses over a positive level, uu, starting from 0, both as uu becomes large and as uu becomes small. Our main focus is on the time, τu\tau_u, it takes the process to transit above the level, and in particular, on the {\it stability} of this passage time; thus, essentially, whether or not τu\tau_u behaves linearly as u\dto 0 or uu\to\infty. We also consider conditional stability of τu\tau_u when the process drifts to -\infty, a.s. This provides information relevant to quantities associated with the ruin of an insurance risk process, which we analyse under a Cram\'er condition

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