First passage upwards for state dependent-killed spectrally negative Lévy processes

Abstract

For a spectrally negative Lévy process (snLp) $X$, killed according to a rate that is a function $\omega$ of its position, we analyse 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 ${\int }_0^{\cdot }\omega (X_u)du$. In particular our findings thus shed extra light on related results concerning first passage times upwards (downwards) of spectrally negative positive self-similar Markov processes (continuous state branching processes).

arXiv comment14 pages, 1 figure