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Hitting probabilities for spectrally Lévy processes. (English) Zbl 0699.60061

Let \(\{\) Y(t), \(t\geq 0\}\) be a spectrally positive Lévy process, that is a process with stationary and independent increments which has no negative jumps, and let q(x) denote the hitting probabilities of the singleton \(\{\) \(x\}\), viz. \(q(x)=\Pr \{Y(t)=x\) for some \(0<t<\infty \}\). A formula for the Laplace transform of q(x) is given, and used to determine explicitly the value of q(0) and \(\lim_{x\to \infty}q(x)\). In the case that \(Y(t)=U(t)-t\), where U is a subordinator, or non-decreasing Lévy process, the first result confirms a recent conjecture of Mallows and Nair and the second extends a result of Nair, Shepp and Klass for the case that U is a Poisson process.
A formula for the Laplace transform of \(E(e^{-\theta T(x)-\sigma D(x)})\) is also given, where T(x) is the time at which the first visit to x takes place, and D(x) is the time between the first passage over x and T(x). A limit theorem for D(x) as \(x\to \infty\) is established and a distributional identity between two path functionals of Brownian motion is derived.

MSC:

60G50 Sums of independent random variables; random walks
60J99 Markov processes
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