Existence of densities for stochastic differential equations driven by Lévy processes with anisotropic jumps

2018/10/17 by Martin Friesen, Peng Jin, Barbara Rüdiger · 1 citation
Mathematics · #math.FA #math.PR #msc:60E07 #msc:60G30 #msc:60H10

paper · pdf · doi:10.1214/20-aihp1077

published as Annales de Institut Henri Poincare - Probabilites et Statistiques (2021), Vol. 57, No. 1

arxiv created 2018/10/17 · arxiv updated 2022/03/17

Abstract

We study existence of densities for solutions to stochastic differential equations with Hölder continuous coefficients and driven by a d-dimensional Lévy process Z=(Z_t)_{t≥ 0}, where, for t>0, the density function f_t of Z_t exists and satisfies, for some (α_i)_i=1,…,d⊂(0,2) and C>0, \limsup _{t → 0}t^1/α_i∫ _ℝ^d|f_t(z+e_ih)-f_t(z)|dz≤ C|h|, h∈ ℝ, i=1,…,d. Here e₁,…,e_d denote the canonical basis vectors in ℝ^d. The latter condition covers anisotropic (α₁,…,α_d)-stable laws but also particular cases of subordinate Brownian motion. To prove our result we use some ideas taken from \citepDF13.

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