Euler scheme for SDEs driven by fractional Brownian motions: Malliavin differentiability and uniform upper-bound estimates

IF 1.1 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications Pub Date : 2024-06-08 DOI:10.1016/j.spa.2024.104412

Jorge A. León , Yanghui Liu , Samy Tindel

引用次数: 0

Abstract

The Malliavin differentiability of a SDE plays a crucial role in the study of density smoothness and ergodicity among others. For Gaussian driven SDEs the differentiability issue is solved essentially in Cass et al., (2013). In this paper, we consider the Malliavin differentiability for the Euler scheme of such SDEs. We will focus on SDEs driven by fractional Brownian motions (fBm), which is a very natural class of Gaussian processes. We derive a uniform (in the step size $n$ ) path-wise upper-bound estimate for the Euler scheme for stochastic differential equations driven by fBm with Hurst parameter $H > 1 / 3$ and its Malliavin derivatives.

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分数布朗运动驱动的 SDE 的欧拉方案：马利亚文可微分性和均匀上界估计

SDE 的马利亚文可微分性在密度平滑性和遍历性等研究中起着至关重要的作用。Cass 等人（2013 年）基本解决了高斯驱动 SDE 的可微分性问题。在本文中，我们将考虑此类 SDE 的欧拉方案的马利亚文可微分性。我们将重点关注由分数布朗运动（fBm）驱动的 SDE，这是一类非常自然的高斯过程。我们推导了一个均匀的（步长为 n 的）路径上界估计值，用于由 Hurst 参数为 H>1/3 的 fBm 驱动的随机微分方程的欧拉方案及其马利亚文导数。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Stochastic Processes and their Applications 数学-统计学与概率论

CiteScore

2.90

自引率

7.10%

发文量

180

审稿时长

23.6 weeks

期刊介绍： Stochastic Processes and their Applications publishes papers on the theory and applications of stochastic processes. It is concerned with concepts and techniques, and is oriented towards a broad spectrum of mathematical, scientific and engineering interests. Characterization, structural properties, inference and control of stochastic processes are covered. The journal is exacting and scholarly in its standards. Every effort is made to promote innovation, vitality, and communication between disciplines. All papers are refereed.