A Ciesielski-Taylor type identity for positive self-similar Markov processes

The aim of this note is to give a straightforward proof of a general version of the Ciesielski-Taylor identity for positive self-similar Markov processes of the spectrally negative type which umbrellas all previously known Ciesielski-Taylor identities within the latter class. The approach makes use of three fundamental features. Firstly, a new transformation which maps a subset of the family of Laplace exponents of spectrally negative Levy processes into itself. Secondly, some classical features of fluctuation theory for spectrally negative Levy processes (see, e.g., [In Seminaire de Probabalites XXXVIII (2005) 16-29 Springer]) as well as more recent fluctuation identities for positive self-similar Markov processes found in [Ann. Inst. H. Poincare Probab. Statist. 45 (2009) 667-684].