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Fluctuation theory and exit systems for positive self-similar Markov processes. (English) Zbl 1241.60019
For a positive self-similar Markov process \(X\), the authors construct a local time for the random set of times for which the process reaches its past supremum. Using this local time the authors describe an exit system for the excursions of \(X\) out of its past supremum. The authors define and study the ladder process \((R,H)\) which is a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time at the past supremum and the process \(X\) sampled on the local time scale. The process \((R,H)\) is described in terms of a ladder process linked to the Lévy process associated to \(X\) via the Lamperti transformation. In the case where \(X\) never hits \(0\), and the upward ladder height is not arithmetic and has finite mean, the authors prove the finite-dimensional convergence of \((R,H)\) as the starting point of \(X\) tends to \(0\). These results are used to provide an alternative proof of the weak convergence of \(X\) as the starting point tends to \(0\). The approach addresses two issues which remained open in [M. E. Caballero and L. Chaumont, Ann. Probab. 34, No. 3, 1012–1034 (2006; Zbl 1098.60038)], namely, how to remove a redundant hypothesis and how to provide a formula for the entrance law of \(X\) in the case where the underlying Lévy process \(X\) oscillates.

MSC:
60G18 Self-similar stochastic processes
60G17 Sample path properties
60J55 Local time and additive functionals
60G51 Processes with independent increments; Lévy processes
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