论文标题
准几何粗糙路径和可变公式的粗略变化
Quasi-geometric rough paths and rough change of variable formula
论文作者
论文摘要
使用Hopf代数和准剃须代数理论中的一些基本概念，我们严格介绍了一个新的粗糙路径的新家族：准几何粗糙路径。我们讨论它们的主要特性。特别是，我们将将它们与迭代的布朗语积分和“简单括号扩展”的概念联系起来，这是在大卫·凯利（David Kelly）的博士学位论文中开发的。由于这些结果，我们有足够的标准可以显示（0,1）$中的任何$γ\以及任何足够光滑的功能$φ\ colon \ colon \ colon \ mathbb {r}^d \ to \ mathbb {r {r} $在任何$γ$-Höldercon$ colon $ x \ colon上的可变公式的粗略变化[r} $ \ mathbb {r}^d $，即在粗糙积分方面的$φ（x_t）$的显式表达式。
Using some basic notions from the theory of Hopf algebras and quasi-shuffle algebras, we introduce rigorously a new family of rough paths: the quasi-geometric rough paths. We discuss their main properties. In particular, we will relate them with iterated Brownian integrals and the concept of "simple bracket extension", developed in the PhD thesis of David Kelly. As a consequence of these results, we have a sufficient criterion to show for any $γ\in (0,1)$ and any sufficiently smooth function $φ\colon \mathbb{R}^d\to \mathbb{R}$ a rough change of variable formula on any $γ$-Hölder continuous path $x\colon [0, T]\to \mathbb{R}^d$, i.e. an explicit expression of $φ(x_t)$ in terms of rough integrals.
