Cramer's estimate for the exponential functional of a Levy process

2002/11/26 by Mejane Olivier, Olivier, Mejane
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.math/0211409

12 pages

arxiv created 2002/11/26 · arxiv updated 2009/11/30

Abstract

We consider the exponential functional A_∞=∫₀^∞ e^ξ_s ds associated to a Levy process (ξ_t)_{t ≥ 0}. We find the asymptotic behavior of the tail of this random variable, under some assumptions on the process ξ, the main one being Cramer's condition, that asserts the existence of a real χ>0 such that \Bbb E(e^χξ₁)=1. Then there exists C>0 satisfying, when t → +∞ : \Bbb P (A_∞> t) ∼ C t^-χ . This result can be applied for example to the process ξ_t = at - S_α(t) where S_α stands for the stable subordinator of index α (0 < α< 1), and a is a positive real (we have then χ=a^1/(α-1)).

Related

• Statistical inference for exponential functionals of Lévy processes
• On the range of exponential functionals of Lévy processes
• Tail asymptotics for exponential functionals of Levy processes: the\n convolution equivalent case
• Cramér asymptotics for finite time first passage probabilities of general Lévy processes
• On the density of exponential functionals of Lévy processes