Path decomposition of a reflected Lévy process on first passage over high levels

Let X be a real valued Lévy process and set R_t=X_t−inf_s≤tX_s. This paper addresses the asymptotic behavior of the sample paths of the reflected process R on first passage over an arbitrarily high level u. We show that under the convolution equivalent condition of Klüppelberg et al. (2004), the sample paths of R on the first excursion which crosses over a high level u can be decomposed into two processes. The first describes the paths in a neighborhood of the origin. The process then takes a large jump into a neighborhood of u. The second process describes the subsequent paths. This sample path behavior is similar to that of X conditioned to cross level u. Using this connection many results concerning, for example, undershoots and overshoots can be easily obtained.