On purely discontinuous additive functionals of subordinate Brownian motions

Abstract

Let A t = s≤t F (X s− , X s ) be a purely discontinuous additive functional of a subordinate Brownian motion X = (X t , P x ). We give a sufficient condition on the non-negative function F that guarantees that finiteness of A ∞ implies finiteness of its expectation. This result is then applied to study the relative entropy of P x and the probability measure induced by a purely discontinuous Girsanov transform of the process X. We prove these results under the weak global scaling condition on the Laplace exponent of the underlying subordinator.