On optimal uniform approximation of Lévy processes on Banach spaces with finite variation processes

For a general \cad Lévy process $X$ on a separable Banach space $V$ we estimate values of $\inf_{c\ge0} \cbr{ \psi(c)+ \inf_{Y\in{\cal A}_{X}(c)}\E \TTV Y{\left[0,T\right]}{}}$, where ${\cal A}_{X}(c)$ is the family of processes on $V$ adapted to the natural filtration of $X$,  a.s. approximating paths of $X$ uniformly with accuracy $c$, $\psi$ is a penalty function with polynomial growth and $\TTV Y{\left[0,T\right]}{}$ denotes the total variation of the process $Y$ on the interval $[0,T]$. Next, we apply obtained estimates in three specific cases: Brownian motion with drift on $\R$, standard Brownian motion on $\R^{d}$ and a symmetric $\alpha$-stable process ($\alpha\in(1,2)$) on $\R$.