A path decomposition for Lévy processes

AbstractExtending a path decomposition which is known to hold both for Brownian motion and random walk, it is shown that an arbitrary oscillatory Lévy process X gives rise to two new independent Lévy processes X(1) and X(2) which have the same law as X and encapsulate the positive (non-positive) excursions of X away from zero, respectively. If X drifts to ±∞, the result also holds with an obvious modification. We discuss various relations between X, X(1) and X(2), but our main focus is on applications. Exploiting the independence of X(1) and X(2) we derive several new distributional results for functionals of X. These include an anlogue for Lévy processes of the well-known fact that the proportion of the time spent in the positive half-line by a Brownian motion with drift before its last visit to zero is uniformly distributed