We consider several first passage problems for stable processes, giving explicitformulas for hitting distributions, hitting probabilities and potentials of stableprocesses killed at first passage. Our principal tools are the Lamperti representationof positive self-similar Markov processes and the Wiener–Hopf factorisationof Lévy processes. As part of the proof apparatus, we introduce a new class ofLévy processes with explicit Wiener–Hopf factorisation, which appear repeatedlyin Lamperti representations derived from stable processes. We also apply theLamperti–Kiu representation of real self-similar Markov processes and obtain resultson the exponential functional of Markov additive processes, in order to findthe law of the first time at which a stable process reaches the origin.
|Date of Award||9 Oct 2013|
|Supervisor||Andreas Kyprianou (Supervisor)|