Markov chain approximations to scale functions of Lévy processes

Abstract

We introduce a general algorithm for the computation of the scale functions of a spectrally negative Lévy 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 its (nonnegative) coefficients given explicitly in terms of the Lévy 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 Itô diffusion setting, where scale functions are expressed in terms of certain integrals of the coefficients of the governing SDE.

46 pages, 4 figures

References in corpus (4)

• On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes
• Markov chain approximations for transition densities of Lévy processes
• Non-random overshoots of Lévy processes
• Fluctuation theory for upwards skip-free Lévy chains

Cited by in corpus (1)

• Fluctuation theory for upwards skip-free Lévy chains