Steven N. Evans (Departments of Statistics and Mathematics, University of California at Be…) Lipschitz minorants of L\'evy processes
The -Lipschitz minorant of a function is the greatest -Lipschitz function dominated pointwise by the function, should such a function exist. I will discuss this construction when the function is a sample path of a (2-sided) L\'evy process. The contact set is the random set of times when the sample path touches the minorant. This is a stationary, regenerative set. I will provide a description of the excursions of the sample path away from the contact set that is analogous to It\^o's theory for the excursions of a Markov process away from some point in the state space. In particular, I will elucidate the probabilistic structure of both a ``generic'' excursion and the special excursion that straddles the time zero.