Markov chain approximations to scale functions of L\'evy processes

TL;DR

This paper presents a stable, easy-to-implement algorithm for approximating the scale functions of spectrally negative Lévy processes using Markov chain discretizations, with proven convergence rates.

Contribution

It introduces a novel algorithm linking Lévy process characteristics to scale functions via Markov chain approximations, with explicit convergence analysis.

Findings

Algorithm is numerically stable and easy to implement.

Provides explicit convergence rates based on process characteristics.

Establishes a clear connection between Lévy triplet and scale functions.

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 its (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.