### Matija Vidmar (University of Warwick)

Title: Markov chain approximations to scale functions of Levy processes

Abstract: We introduce a general algorithm for the computation of the scale functions of a spectrally negative Levy process X, based on a natural weak approximation of X via upwards skip-free continuous-time Markov chains with stationary independent increments. The algorithm consists of evaluating a finite linear recursion with coefficients given explicitly in terms of the Levy triplet of X. It is easy to implement and fast to execute. Our main result establishes sharp rates of convergence of this algorithm providing an explicit link between the semimartingale characteristics of X and its scale functions, not unlike the one-dimensional Ito diffusion setting, where scale functions are expressed in terms of certain integrals of the coefficients of the governing SDE.