L\'evy-type processes: convergence and discrete schemes

TL;DR

This paper characterizes the convergence of discrete Markov processes to locally Feller processes, applying the theory to Levy-type processes, diffusions, and random walks in random media, with implications for simulation and approximation methods.

Contribution

It introduces a unified framework for analyzing convergence of discrete schemes to complex continuous processes, including Levy-type processes and diffusions in random environments.

Findings

Discrete Markov processes converge to locally Feller processes under operator convergence.

Euler schemes effectively approximate Levy-type processes and singular diffusions.

Random walks in random media converge to diffusions in random potentials.

Abstract

We characterise the convergence of a certain class of discrete time Markov processes toward locally Feller processes in terms of convergence of associated operators. The theory of locally Feller processes is applied to L\'evy-type processes in order to obtain convergence results on discrete and continuous time indexed processes, simulation methods and Euler schemes. We also apply the same theory to a slightly different situation, in order to get results of convergence of diffusions or random walks toward singular diffusions. As a consequence we deduce the convergence of random walks in random medium toward diffusions in random potential.