Moment determinacy of the reciprocal exponential functional (Bertoin–Yor conjecture)

Determine whether, for an unkilled Lévy process whose exponential moments are finite of all positive orders (equivalently, c^Ψ = −∞), the reciprocal of its exponential functional I_Ψ = ∫_0^∞ e^{−ξ_s} ds is moment determinate if and only if the Lévy measure Π has no mass on (0, ∞) (i.e., the process has no positive jumps).

Background

In the regime c^Ψ = −∞ (all positive exponential moments finite), the exponential functional I_Ψ has all negative moments. Bertoin and Yor proposed a sharp criterion for moment determinacy of the reciprocal random variable I_Ψ^{-1}: it should be moment determinate precisely when the underlying Lévy process has no positive jumps.

The survey recalls explicit formulas for negative moments and discusses why standard determinacy conditions (e.g., Carleman) may fail when Π((0,∞)) > 0, motivating the conjecture and its unresolved status.