On Some Properties of the Fractional Derivative of the Brownian Local Time

Pub Date : 2024-07-11 DOI:10.1134/s0081543824010115

I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev

引用次数: 0

Abstract

We study the properties of the fractional derivative \(D_\alpha l(t,x)\) of order \(\alpha<1/2\) of the Brownian local time \(l(t,x)\) with respect to the variable \(x\). This derivative is understood as the convolution of the local time with the generalized function \(|x|^{-1-\alpha}\). We show that \(D_\alpha l(t,x)\) appears naturally in Itô’s formula for the process \(|w(t)|^{1-\alpha}\). Using the martingale technique, we also study the limit behavior of \(D_\alpha l(t,x)\) as \(t\to\infty\).

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论布朗局部时间分数衍生物的一些特性

摘要我们研究了布朗局部时间\(l(t,x)\)相对于变量\(x\)的阶\(\alpha<1/2\)的分数导数\(D_\alpha l(t,x)\)的性质。这个导数可以理解为局部时间与广义函数 \(|x|^{-1-\alpha}\)的卷积。我们证明，\(D_\alpha l(t,x)\) 会自然地出现在伊托过程公式中\(|w(t)|^{1-\alpha}\)。利用马丁格尔技术，我们还研究了 \(t\to\infty\) 时 \(D_α l(t,x)\) 的极限行为。

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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