Moments of passage times for Lévy processes. (English) Zbl 1042.60025

Doney, R. A.; Maller, R. A.

Ann. Inst. Henri Poincaré, Probab. Stat. 40, No. 3, 279-297 (2004).

Summary: The authors give necessary and sufficient conditions, in terms of characteristics of the process, for finiteness of moments of passage times of general Lévy processes above horizontal, linear or certain curved boundaries. They apply in particular to processes which drift almost surely

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to infinity, and lead to estimates of the rate of growth of certain expectations, constituting generalised kinds of renewal theorems. Further results concern the inverse local time at the maximum and the ladder height process, the amount of time spent below a given level, and the overall minimum of the Lévy process.

MSC:

60G51	Processes with independent increments; Lévy processes
60K05	Renewal theory
60F15	Strong limit theorems
60G40	Stopping times; optimal stopping problems; gambling theory
60G10	Stationary stochastic processes
60G17	Sample path properties
60J65	Brownian motion
60G50	Sums of independent random variables; random walks

Keywords:

Passage times of Lévy processes; Horizontal boundaries; Curved boundaries; Drift to infinity; Renewal theorems; Local time; Ladder height processes; Strong laws