Noise reinforced Lévy processes: Lévy-Itô decomposition and applications

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 Lévy process for a time partition of mesh-size $1∕n$ converges, as $n\uparrow \mathrm{\infty }$ in the sense of finite dimensional distributions, to a process $\hat{\mathrm{\xi }}$ referred to as a noise reinforced Lévy process. Our first main result states that a noise reinforced Lévy process has rcll paths and satisfies a $\text{noise reinforced}$ Lévy-Itô decomposition in terms of the $\text{noise reinforced}$ Poisson point process of its jumps. We introduce the joint distribution of a Lévy process and its reinforced version $(\mathrm{\xi },\hat{\mathrm{\xi }})$ and show that the pair, conformed by the skeleton of the Lévy process and its step reinforced version, converge towards $(\mathrm{\xi },\hat{\mathrm{\xi }})$ as the mesh size tend to 0. As an application, we analyse the rate of growth of $\hat{\mathrm{\xi }}$ at the origin and identify its main features as an infinitely divisible process.