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On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. (English) Zbl 1049.60042
Let $$X$$ be a spectrally one-sided Lévy process and reflect it at its past infimum $$I$$. That is the Lévy process minus its past infimum $$Y=X-I$$. The author studies such a process. Here he determines the Laplace transform for $$Y$$ if $$X$$ is spectrally negative. Subsequently, he finds an expression for the resolvent measure for $$Y$$ killed upon leaving $$[0,a]$$ and studies some properties of this process. Finally, the rate of convergence of the supremum of the reflected process to $$a$$ is given.

##### MSC:
 60G51 Processes with independent increments; Lévy processes 60G70 Extreme value theory; extremal stochastic processes
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