engleski

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 $X=(X_t, \P_x)$. We give a sufficient condition on the non-negative function $F$ that guarantees that finiteness of $A_{; ; ; \infty}; ; ; $ 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.

Additive functionals, subordinate Brownian motion, purely discontinuous Girsanov transform, absolute continuity, singularity, relative entropy.

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