Numerical Techniques in Lévy Fluctuation Theory

This paper presents a framework for numerical computations in fluctuation theory for Lévy processes. More specifically, with X¯t:=sup0≤s≤tXs denoting the running maximum of the Lévy process X t , the aim is to evaluate P(X¯t≤x) for t,x > 0. We do so by approximating the Lévy process under consideration by another Lévy process for which the double transform Ee−αX¯τ(q) is known, with τ(q) an exponentially distributed random variable with mean 1/q; then we use a fast and highly accurate Laplace inversion technique (of almost machine precision) to obtain the distribution of X¯t . A broad range of examples illustrates the attractive features of our approach.