For a general càdlàg Lévy process on a separable Banach space $V$ we estimate values of $\inf_{Y\in{\cal A}_X} \mathbb{E}\left\{ ψ\left( \Vert X - Y \Vert_\infty\right) + \mathrm{TV}(Y[0,T]) \right\}$, where ${\cal A}_X$ is the family of processes on $V$ adapted to the natural filtration of $X$, $ψ$ 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 $α$-stable process ($α\in(1,2)$) on $\mathbb{R}$.