Correlated continuous time random walks and fractional Pearson diffusions

TL;DR

This paper introduces correlated continuous time random walks that converge to fractional Pearson diffusions, using Markov chain-based jumps and stable law waiting times, resulting in non-Lévy diffusion processes with dependent increments.

Contribution

It develops a novel class of correlated CTRWs converging to fractional Pearson diffusions, incorporating Markov chain jumps and stable law waiting times.

Findings

The correlated CTRWs converge to fractional Pearson diffusions.

Jumps are modeled via Bernoulli urn-scheme and Wright-Fisher models.

Limiting processes exhibit non-independent increments.

Abstract

Continuous time random walks have random waiting times between particle jumps. We define the correlated continuous time random walks (CTRWs) that converge to fractional Pearson diffusions (fPDs). The jumps in these CTRWs are obtained from Markov chains through the Bernoulli urn-scheme model and Wright-Fisher model. The jumps are correlated so that the limiting processes are not L\'evy but diffusion processes with non-independent increments. The waiting times are selected from the domain of attraction of a stable law