Tail-equivalence conjecture for exponential functionals (Rivero’s conjecture)

Establish that the tail of the exponential functional I_Ψ = ∫_0^∞ e^{−ξ_s} ds satisfies F_Ψ(x) = P(I_Ψ > x) ∼ c P(e^{−inf_{s≥0} ξ_s} > x) as x → ∞ for some constant c > 0, for general Lévy processes beyond currently proved subclasses.

Background

A three-factor decomposition I_Ψ ≍ I_{φ+} × Y{φ-}^{−1} × e^{−inf{s≥0} ξs} suggests that the upper tail of IΨ should be governed by e^{−inf ξ}. Rivero conjectured a precise tail equivalence with an explicit constant.

The survey presents results confirming this equivalence under Cramér-type and convolution-equivalent assumptions, but the general validity is not proved and remains conjectural.