On fluctuation-theoretic decompositions via Lindley-type recursions

Consider a Lévy process Y (t) over an exponentially distributed time
with mean T_ß  . We study the joint distribution of the  unning
 maximum  Yand the time epoch G  (T_ß ) at which this maximum last occurs. Our main result is a fluctuation-theoretic distributional equality: the vector (Y (T_ß ), G  (T_ß )) can be written  as a sum o_f two independent vectors, the first one being (Y (T_ß+ω), G (T_ß+ω) and the second one being the running maximum and corresponding time epoch under the restriction that the Lévy process is only observed at Poisson (ω) inspection epochs (until T_ß). We first provide an analytic proof for this remarkable decomposition, and then a more elementary proof that gives insight into the occurrence of the decomposition and into the fact that
only appears in the right hand side of the decomposition. The proof technique underlying the more elementary derivation also leads to further generalizations of the decomposition, and to some fundamental insights into a generalization of the well known Lindley recursion.