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

2018/08/25 by W. M. Bednorz, Bednorz, W. M., Rafał M. Łochowski +3
Economics, Econometrics and Finance · Mathematics · #Advanced Banach Space Theory #Nonlinear Differential Equations Analysis #Stochastic processes and financial applications #math.PR #msc:60G51

paper · pdf · doi:10.48550/arxiv.1808.08373

arxiv created 2020/09/30 · arxiv updated 2020/10/01

Abstract

For a general càdlàg Lévy process on a separable Banach space V we estimate values of inf_Y∈\cal A_X 𝔼\ ψ( \Vert X - Y \Vert_∞) + TV(Y[0,T]) \, 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 ℝ, a standard Brownian motion on ℝ^d and a symmetric α-stable process (α∈(1,2)) on ℝ.

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