Discrete random walk models for symmetric Levy-Feller diffusion processes

TL;DR

This paper introduces discrete random walk models that accurately simulate symmetric Levy-Feller diffusion processes, demonstrating convergence to continuous models as space and time steps diminish.

Contribution

The authors develop new discrete models for symmetric Levy-Feller processes that converge to the continuous processes, extending simulation capabilities for arbitrary stability indices.

Findings

Models successfully simulate symmetric Levy-Feller processes.

Proper scaling ensures convergence to continuous diffusion processes.

Applicable for arbitrary stability index $eta$ in (0, 2].

Abstract

We propose a variety of models of random walk, discrete in space and time, suitable for simulating stable random variables of arbitrary index $\alpha$ ($0< \alpha \le 2$), in the symmetric case. We show that by properly scaled transition to vanishing space and time steps our random walk models converge to the corresponding continuous Markovian stochastic processes, that we refer to as Levy-Feller diffusion processes.