On a moment determinacy conjecture of Bertoin and Yor

Abstract

Let $\xi$ be an unkilled real-valued Lévy process which drifts to $+\infty$ and has positive exponential moments of all orders, and define $I_{\xi }={\int }_0^{\infty }{e}^{-{\xi }_t},dt$, and its reciprocal $X_{\xi }=1/I_{\xi }$. Bertoin and Yor proved that $X_{\xi }$ is moment-determinate when $\xi$ has no positive jumps, and conjectured that this condition is also necessary. We prove the latter. The proof is based on a lower bound near zero for the law of $I_{\xi }$. We show that a group of sufficiently many positive jumps near the origin puts $I_{\xi }$ on a suitable small scale. The first selected jump time is used as a one-dimensional smooth coordinate, yielding an absolutely continuous subcomponent of the law of $I_{\xi }$. After the change of variables, the resulting subdensity of $X_{\xi }$ satisfies a Krein moment indeterminacy criterion.