On Exit and Ergodicity of the Spectrally One-Sided Lévy Process Reflected at Its Infimum

M. R. Pistorius ()
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M. R. Pistorius: King's College London, Strand

Journal of Theoretical Probability, 2004, vol. 17, issue 1, 183-220

Abstract: Abstract Consider a spectrally one-sided Lévy process X and reflect it at its past infimum I. Call this process Y. For spectrally positive X, Avram et al.(2) found an explicit expression for the law of the first time that Y=X−I crosses a finite positive level a. Here we determine the Laplace transform of this crossing time for Y, if X is spectrally negative. Subsequently, we find an expression for the resolvent measure for Y killed upon leaving [0,a]. We determine the exponential decay parameter ϱ for the transition probabilities of Y killed upon leaving [0,a], prove that this killed process is ϱ-positive and specify the ϱ-invariant function and measure. Restricting ourselves to the case where X has absolutely continuous transition probabilities, we also find the quasi-stationary distribution of this killed process. We construct then the process Y confined in [0,a] and prove some properties of this process.

Keywords: Reflected Lévy process; first passage; exponential decay; excursion theory; h-transform (search for similar items in EconPapers)
Date: 2004
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Citations: View citations in EconPapers (3)

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DOI: 10.1023/B:JOTP.0000020481.14371.37

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