Asymptotic Estimation of Finite Horizon Ruin Probability for Random Walks with Heavy Tailed Increments ThroughCorrected Diffusion Approximations

Abstract

Diffusion approximation has been established as an efficient tool for estimating ruin probability, i.e. first passage probability, for very general random walks. Especially, when the so called ”heavy traffic conditions” is satisfied, the estimation could be very accurate. Efforts have been made for augmenting this approach to improve its accuracy for the cases that the ”heavy traffic conditions” are not satisfied. Most notably, D. Siegmund (Siegmund, 1979) and M. Hogan (Hogan, 1986) develop corrected diffusion approximations to estimate ruin probabilities for infinite horizon and finite horizon with light tail increments. Recently, Asmussen (Asmussen, 2000) and his colleagues extend corrected diffusion approximation for random summations. In this note, we extend the methodology used in M. Hogan (Hogan, 1986), which does not require the existence of the moment generating function for the increment of the random walk, to the case of finite horizon. Replacing the fixed time epoch, we will use a Poisson process with rate 1 as a random clock, thus transform the problem into the studying of the passage time for a special two dimensional random walk. Conducting the similar Fourier analysis as those in (Hogan, 1986), we are able to obtain the asymptotic estimation in integration forms.