Uniform asymptotics of ruin probabilities for L´ evy processes M. Kelbert ∗ , I.Sazonov † and F.Avram ‡ Key words: Ruin probability, Cram´ er-Lundberg model, L´ evy process, Cre- mona equation, Saddle-point approximation, Fresnel integral AMS Classification 2010: 60G51, 62P05 Abstract In this paper we obtain, for spectrally negative L´ evy processes X, uniform approximations for the finite time ruin probability Ψ(t, u)= P u [T ≤ t],T = inf {t ≥ 0: X (t) < 0}, when u = X (0) and t tend to infinity such that v = u/t is constant, and the so-called Cram´ er light-tail condition is satisfied. 1 Introduction This paper is motivated by Arfwedson’s [1] saddle-point approximation for the finite time ruin probabilities of the Cram´ er-Lundberg ruin model. His approach treats two distinct cases of “quick and slow ruin” depending the value of parameter v = u/t where u is the initial capital, but does not cover the Stokes line u/t = v cr separating them. As known since Van der Waerden [15], this “non-uniformity” is a typical problem arising by applying the classical SP (saddle-point) approximation of Laplace-type integrals when * Dept of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, UK. Institute of Earthquake Prediction Theory and Mathematical Geophysics RAS, m.kelbert@swansea.ac.uk † College of Engineering, Swansea University, Singleton Park, Swansea, SA2 8PP, UK. Institute of Atmospheric Physics RAS, i.sazonov@swansea.ac.uk ‡ Dept de Math´ emathiques, Universit´ e de Pau, France, Florin.Avram@univ-Pau.fr 1