Noise Reinforced Lévy Processes: Lévy-Itô Decomposition and Applications
英文摘要：
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 $n \uparrow \infty$ in the sense of finite dimensional distributions, to a process $\hatξ$ referred to as a noise reinforced Lévy process. Our first main result states that a noise reinforced Lévy processes has rcll paths and satisfies a $\textit{noise reinforced}$ Lévy Itô decomposition in terms of the $\textit{noise reinforced}$ Poisson point process of its jumps. We introduce the joint distribution of a Lévy process and its reinforced version $(ξ, \hatξ)$ and show that the pair, conformed by the skeleton of the Lévy process and its step reinforced version, converge towards $(ξ, \hatξ)$ as the mesh size tend to $0$. As an application, we analyse the rate of growth of $\hatξ$ at the origin and identify its main features as an infinitely divisible process.
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