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Passage Time and Fluctuation Calculations for Subexponential Levy Processes

Passage Time and Fluctuation Calculations

Ron Doney, Claudia Kluppelberg and Ross Maller

We consider the passage time problem for L'evy processes, emphasising heavy tailed cases, with a view to applications in insurance risk. Results are obtained under quite mild assumptions, namely, drift to *minus infinity a.s. of the process, possibly at a linear rate (the finite mean case), but possibly much faster (the infinite mean case), together with subexponential growth.

Local, and functional, versions of limit distributions are derived for the passage time itself, as well as for the position of the process just prior to passage, and the overshoot of a high level.  

A significant connection is made with extreme value theory via regular variation or maximum domain of attraction conditions imposed on the positive tail of the canonical measure, which are shown to be necessary for the kind of convergence behaviour we are interested in. Specialisation of the L'evy results to random walk situations is outlined.

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