Hölder regularity and gradient estimates Hölder regularity and gradient estimates for SDEs driven by cylindrical α-stable processes

2020/01/12 by Zhen-Qing Chen, Chen, Zhen-Qing, Zimo Hao +3
Computer Science · Economics, Econometrics and Finance · Mathematics · #60H10 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2001.03873

openalex publication_date 2020/01/12 · openalex created_date 2020/01/23 · openalex updated_date 2026/07/28

Abstract

We establish Hölder regularity and gradient estimates for the transition semigroup of the solutions to the following SDE: \rm d X_t=σ(t, X_t-)\rm d Z_t+b (t, X_t)\rm d t, X₀=x∈\mathbb R^d, where ( Z_t)_{t≥ 0} is a d-dimensional cylindrical α-stable process with α∈ (0, 2), σ(t, x):\mathbb R₊×\mathbb R^d→\mathbb R^d⊗\mathbb R^d is bounded measurable, uniformly nondegenerate and Lipschitz continuous in x uniformly in t, and b (t, x):\mathbb R₊×\mathbb R^d→\mathbb R^d is bounded β-Hölder continuous in x uniformly in t with β∈[0,1] satisfying α+β>1. Moreover, we also show the existence and regularity of the distributional density of X (t, x). Our proof is based on Littlewood-Paley's theory.

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