混合粗糙微分方程的精确拉普拉斯近似

IF 2.4 2区 数学 Q1 MATHEMATICS
Xiaoyu Yang , Yong Xu , Bin Pei
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引用次数: 0

摘要

本文主要研究了混合粗糙路径(BH,W)驱动的粗糙微分方程(RDE)的拉普拉斯近似,H∈(1/3,1/2)为ε→0。首先,基于从混合分数布朗运动(fBm)推导出的几何粗糙路径,给出了 RDE 解的第一级路径规律的 Schilder 型大偏差原理(LDP)。由于混合粗糙路径的特殊性,进行拉普拉斯近似的主要困难在于证明限制在混合 fBm 的 Cameron-Martin 空间上的 Itô 映射的 Hessian 矩阵的 Hilbert-Schmidt 属性。为此,我们将卡梅隆-马丁空间嵌入到一个更大的希尔伯特空间中,那么赫希矩阵就是可计算的。随后,我们展示了 Hessian 的概率表示。最后,我们构建了拉普拉斯近似值,从而得出了指数尺度下更精确的渐近线。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Precise Laplace approximation for mixed rough differential equation

This work focuses on the Laplace approximation for the rough differential equation (RDE) driven by mixed rough path (BH,W) with H(1/3,1/2) as ε0. Firstly, based on geometric rough path lifted from mixed fractional Brownian motion (fBm), the Schilder-type large deviation principle (LDP) for the law of the first level path of the solution to the RDE is given. Due to the particularity of mixed rough path, the main difficulty in carrying out the Laplace approximation is to prove the Hilbert-Schmidt property for the Hessian matrix of the Itô map restricted on the Cameron-Martin space of the mixed fBm. To this end, we embed the Cameron-Martin space into a larger Hilbert space, then the Hessian is computable. Subsequently, the probability representation for the Hessian is shown. Finally, the Laplace approximation is constructed, which asserts the more precise asymptotics in the exponential scale.

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来源期刊
CiteScore
4.40
自引率
8.30%
发文量
543
审稿时长
9 months
期刊介绍: The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools. Research Areas Include: • Mathematical control theory • Ordinary differential equations • Partial differential equations • Stochastic differential equations • Topological dynamics • Related topics
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