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First passage upwards for state-dependent-killed spectrally negative Lévy processes. (English) Zbl 1415.60050

Summary: For a spectrally negative Lévy process \(X\), killed according to a rate that is a function \(\omega\) of its position, we complement the recent findings of B. Li and Z. Palmowski [Stochastic Processes Appl. 128, No. 10, 3273–3299 (2018; Zbl 1401.60087)] by analysing (in greater generality) the exit probability of the one-sided upwards passage problem. When \(\omega\) is strictly positive, this problem is related to the determination of the Laplace transform of the first passage time upwards for \(X\) that has been time-changed by the inverse of the additive functional \(\int_0^\cdot \omega (X_u)\,\mathrm{d}u\). In particular, our findings thus shed extra light on related results concerning first passage times downwards (resp. upwards) of continuous-state branching processes (resp. spectrally negative positive self-similar Markov processes).

MSC:

60G51 Processes with independent increments; Lévy processes
60J25 Continuous-time Markov processes on general state spaces
60G44 Martingales with continuous parameter

Citations:

Zbl 1401.60087
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References:

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