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

¹ University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
² Warsaw School of Economics, al. Niepodległości 162, 02-554 Warszawa, Poland

^* Corresponding author: rlocho@sgh.waw.pl

Received: 13 October 2020
Accepted: 27 October 2022

Abstract

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) 𝔼TV(Y,[0,T])}, where 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, 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 ℝ^d and a symmetric α-stable process (α ∈ (1, 2)) on ℝ.