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

For a general càdlàg Lévy process X on a separable Banach space V we estimate values of inf_{c≥0} {ψ(c) + inf_{Y∈AX(c)} TV(Y,[0,T])}, where AX(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, ψ is a penalty function with 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: Brownian motion with drift on ℝ, standard Brownian motion on ℝd and a symmetric α-stable process (α ∈ (1, 2)) on ℝ.