On purely discontinuous additive functionals of subordinate Brownian motions

Let $A_t={\sum }_{s\le t}F(X_{s-},X_s)$ be a purely discontinuous additive functional of a subordinate Brownian motion $X=(X_t,{\mathbb{P}}_x)$. We give a sufficient condition on the non-negative function $F$ that guarantees that finiteness of $A_{\mathrm{\infty }}$ implies finiteness of its expectation. This result is then applied to study the relative entropy of ${\mathbb{P}}_x$ 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.