ABSTRACT

In this article, we will present a new perspective on the variable-order fractional calculus, which allows for differentiation and integration to a variable order. The concept of multifractional calculus has been a scarcely studied topic within the field of functional analysis in the past 20 years. We develop a multifractional differential operator which acts as the inverse of the multifractional integral operator. This is done by solving the Abel integral equation generalized to a multifractional order. With this new multifractional differential operator, we prove a Girsanov's theorem for multifractional Brownian motions of Riemann–Liouville type. As an application, we show how Girsanov's theorem can then be applied to prove the existence of a unique strong solution to stochastic differential equations where the drift coefficient is merely of linear growth, and the driving noise is given by a non-stationary multifractional Brownian motion with a Hurst parameter as a function of time. The Hurst functions we study will take values in a bounded subset of $(0,\frac{1}{2})$. The application of multifractional calculus to SDEs is based on a generalization of the works of D. Nualart and Y. Ouknine [Regularization of differential equations by fractional noise, Stoch Process Appl. 102(1) (2002), pp. 103–116].

摘要

在本文中，我们将介绍变阶分数微积分的新观点，它允许微分和积分到变阶。在过去的 20 年中，多分数微积分的概念一直是泛函分析领域中很少研究的话题。我们开发了一个多分数微分算子，它作为多分数积分算子的逆。这是通过求解泛化为多分数阶的 Abel 积分方程来完成的。使用这种新的多分数微分算子，我们证明了黎曼-刘维尔型多分数布朗运动的 Girsanov 定理。作为一个应用程序，我们展示了 Girsanov' 然后可以应用 s 定理来证明随机微分方程的唯一强解的存在，其中漂移系数仅线性增长，并且驱动噪声由具有赫斯特参数作为函数的非平稳多分数布朗运动给出的时间。我们研究的 Hurst 函数将在有界子集中取值$(0,\frac{1}{2})$. 多元微积分在 SDE 中的应用是基于对 D. Nualart 和 Y. Ouknine 工作的概括[通过分数噪声对微分方程进行正则化，Stoch Process Appl。102(1) (2002)，第 103-116 页]。