On purely discontinuous additive functionals of subordinate Brownian motions

TL;DR

This paper establishes conditions under which the finiteness of a purely discontinuous additive functional of a subordinate Brownian motion implies the finiteness of its expectation, with applications to entropy and measure transforms.

Contribution

It provides a sufficient condition linking the finiteness of an additive functional to its expectation for subordinate Brownian motions, under weak scaling assumptions.

Findings

Finiteness of $A_$ implies finiteness of expectation under certain conditions.

Application to relative entropy between probability measures.

Analysis of Girsanov transforms for subordinate Brownian motions.

Abstract

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 $A_{\infty}$ 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.