On a moment determinacy conjecture of Bertoin and Yor

Abstract

Let $ξ$ be an unkilled real-valued Lévy process which drifts to $+\infty$ and has positive exponential moments of all orders, and define $I_ξ=\int_0^\infty e^{-ξ_t},dt$, and its reciprocal $X_ξ=1/I_ξ$. Bertoin and Yor proved that $X_ξ$ is moment-determinate when $ξ$ 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 $I_ξ$. We show that a group of sufficiently many positive jumps near the origin puts $I_ξ$ 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 $I_ξ$. After the change of variables, the resulting subdensity of $X_ξ$ satisfies a Krein moment indeterminacy criterion.