Rough path 理论是由 Terry Lyons 在97年提出的一套定义和分析连续的粗糙（无界变差）的轨道所驱动的积分和方程的理论方法。它的重要意义在于给出了由轨道粗糙的随机过程（如布朗运动）所驱动的积分和方程的一种按轨道定义它们的方法。近几年，rough path理论被广泛的应用于随机偏微分方程，随机动力系统，以及机器学习等领域中。此前关于rough path理论的研究，大部分是针对于连续的情况。这主要是由于之前并没有一个完整的允许带跳的理论框架，因此rough path理论在更一般的带跳的情况下的应用有些捉襟见肘。本项目就是针对允许带跳情况下的rough path理论，完善本人之前与合作者建立的允许带跳的 rough path的理论结果。并且更进一步的探索该理论在随机偏微分方程，如由levy噪声驱动的随机偏微分方程，以及在机器学习等方面的应用。

Rough path theory is built by Terry Lyons in his seminal work in 1997. It is used to describe integration and differential equations driven by paths with unbounded variation. The main application of this theory is to describe integration and differential equations driven by processes with rough sample paths. In recent years, this theory is widely applied in areas such as stochastic partial differential equations, stochastic dynamic systems, and machine learning. Most earlier work about rough paths is based on continuous framework. The main reason is the lack of a framework allowing jumps, so when facing with the jump case, this theory is sometimes powerless. This project is aimed at completing the general rough path framework allowing jumps built by us, and further developing more applications in stochastic partial differential equations driven by levy kind of noise and machine learning.