A Refined Factorization of the Exponential Law

Let ξ be a (possibly killed) subordinator with Laplace exponent φ and denote by Iφ = ∫∞ 0 e−ξsds, the so-called exponential functional. Consider the positive random variable Iψ1 whose law, according to Bertoin and Yor [6], is determined by its negative entire moments as follows E[I ψ1 ] = n ∏ k=1 φ(k), n = 1, 2 . . . . In this note, we show that Iψ1 is a positive self-decomposable random variable whenever the Lévy measure of ξ is absolutely continuous with a monotone decreasing density. In fact, Iψ1 is identified as the exponential functional of a spectrally negative (for short sn) Lévy process. We deduce from [6] the following factorization of the exponential law e Iφ/Iψ1 (d) = e where Iψ1 is taken independent of Iφ. We proceed by showing an identity in distribution between the entrance law of a sn self-similar positive Feller process and the reciprocal of the exponential functional of sn Lévy processes. As by-product, we get, some new examples of the law of the exponential functionals, a new factorization of the exponential law and some interesting distributional properties of some random variables. For instance, we obtain that S(α) is a self-decomposable random variable where S(α) is a positive stable random variable of index α ∈ (0, 1).