Optimal uniform approximation of Lévy processes on Banach spaces with finite variation processes

Abstract

For a general càdlàg Lévy process on a separable Banach space $V$ we estimate values of ${\inf }_{Y\in {\mathcal{A}}_X}\mathbb{E}\left\{\psi \left(\Vert X-Y{\Vert }_{\infty }\right)+\mathrm{TV}(Y[0,T])\right\}$, where ${\mathcal{A}}_X$ 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 $\mathbb{R}$.