Noise Reinforced L\'evy Processes: L\'evy-It\^o Decomposition and Applications

TL;DR

This paper introduces noise reinforced Lévy processes, proves their path properties and Lévy-Itô decomposition, and explores their convergence from discrete reinforced random walks, highlighting their growth behavior and infinite divisibility.

Contribution

It establishes the Lévy-Itô decomposition for noise reinforced Lévy processes and demonstrates their convergence from discrete reinforced random walks, extending Lévy process theory.

Findings

Noise reinforced Lévy processes have right-continuous with left limits paths.

They satisfy a Lévy-Itô decomposition involving reinforced Poisson jumps.

The reinforced processes converge from discrete step reinforced random walks as mesh size decreases.

Abstract

A step reinforced random walk is a discrete time process with memory such that at each time step, with fixed probability $p \in (0,1)$, it repeats a previously performed step chosen uniformly at random while with complementary probability $1-p$, it performs an independent step with fixed law. In the continuum, the main result of Bertoin in [7] states that the random walk constructed from the discrete-time skeleton of a L\'evy process for a time partition of mesh-size $1/n$ converges, as $n \uparrow \infty$ in the sense of finite dimensional distributions, to a process $\hat{\xi}$ referred to as a noise reinforced L\'evy process. Our first main result states that a noise reinforced L\'evy processes has rcll paths and satisfies a $\textit{noise reinforced}$ L\'evy It\^o decomposition in terms of the $\textit{noise reinforced}$ Poisson point process of its jumps. We introduce the joint distribution of a L\'evy process and its reinforced version $(\xi, \hat{\xi})$ and show that the pair, conformed by the skeleton of the L\'evy process and its step reinforced version, converge towards $(\xi, \hat{\xi})$ as the mesh size tend to $0$. As an application, we analyse the rate of growth of $\hat{\xi}$ at the origin and identify its main features as an infinitely divisible process.