Optimal uniform approximation of L\'evy processes on Banach spaces with finite variation processes

TL;DR

This paper provides optimal uniform approximation estimates for Lévy processes on Banach spaces using finite variation processes, with applications to Brownian motion and stable processes.

Contribution

It introduces a general framework for approximating Lévy processes with finite variation processes and derives explicit bounds in key cases.

Findings

Derived bounds for approximation errors of Lévy processes

Applied estimates to Brownian motion and stable processes

Demonstrated effectiveness of approximation methods

Abstract

For a general c\`adl\`ag L\'evy process on a separable Banach space $V$ we estimate values of $\inf_{Y\in{\cal A}_X} \mathbb{E}\left\{ \psi\left( \Vert X - Y \Vert_\infty\right) + \mathrm{TV}(Y[0,T]) \right\}$, where ${\cal A}_X$ is the family of processes on $V$ adapted to the natural filtration of $X$, $\psi$ has polynomial growth and TV$(Y[0,T])$ denotes the total variation of the process $Y$ on the interval $[0,T]$. Next, we apply obtained estimates in three specific cases: a Brownian motion with drift on $\mathbb{R}$, a standard Brownian motion on $\mathbb{R}^d$ and a symmetric $\alpha$-stable process ($\alpha\in(1,2)$) on $\mathbb{R}$.