Multiplicative L\'evy processes: It\^o versus Stratonovich interpretation

TL;DR

This paper compares Itô and Stratonovich interpretations of Langevin equations with multiplicative Lévy noise, analyzing their effects on diffusion properties and providing analytical and numerical insights.

Contribution

It introduces a detailed comparison of Itô and Stratonovich interpretations for Lévy-driven Langevin equations, highlighting differences in diffusion behavior and variance.

Findings

Variance is infinite in Itô case

Variance can be finite and subdiffusive in Stratonovich case

Analytical results align with numerical simulations

Abstract

Langevin equation with a multiplicative stochastic force is considered. That force is uncorrelated, it has the L\'evy distribution and the power-law intensity. The Fokker-Planck equations, which correspond both to the It\^o and Stratonovich interpretation of the stochastic integral, are presented. They are solved for the case without drift and for the harmonic oscillator potential. The variance is evaluated; it is always infinite for the It\^o case whereas for the Stratonovich one it can be finite and rise with time slower that linearly, which indicates subdiffusion. Analytical results are compared with numerical simulations.