Stable processes with an absorbing barrier

First passage times for symmetric stable processes in space

“…The right continuity of the paths implies that T is a random variable. The fact that the process under consideration has the extended Markov property implies at once that the following first passage relation holds (this is often called the Désiré-André equation; see [12])…”

On the distribution of first hits for the symmetric stable processes.

“…All the information necessary to determine the hitting probabilities ux and u* is contained in the "first passage time relation" or "Désiré-André equation" (cf. [6]). Let A be an open set in £", A its closure.…”

On the scaling property in fluctuation theory for stable Lévy processes

“…This law is already known. In [8] Ray gives an expression of its density in the symmetric stable case. In [2] Bingham generalises this result for non-spectrally negative stable processes.…”

“…In [8], Ray gives an expression for the density of the law of X T [x,+∞[ in the symmetric case. In [2], Bingham generalizes this result to the case when X is not spectrally negative.…”