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 L\'evy 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, what are nonnegative, coefficients given explicitly in terms of the L\'evy triplet of $X$. Thus it is easy to implement and numerically stable. 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 It\^o diffusion setting, where scale functions are expressed in terms of certain integrals of the coefficients of the governing SDE.

Matlab code for the algorithm for computing scale functions of spectrally negative Levy processes based on this paper is available here.

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