论文标题
固有的随机微分方程并扩展了歧管上的ITO公式
Intrinsic Stochastic Differential Equations and Extended Ito Formula on Manifolds
论文作者
论文摘要
在平滑歧管上代表随机微分方程（SDE）的一种一般方法是基于施瓦茨的形态。在此手稿中，我们对由p维wiener流程驱动$ w_t \ in \ mathbb {r}^p $驱动的平滑歧管$ m $上的SDE感兴趣。就施瓦茨的形态而言，这种SDEs由Schwartz Morphism表示，该形态会在\ Mathbb {r}^{r}^{p+1} $中变形到歧管上的半木制中。我们表明，可以使用我们称为扩散发生器的特殊地图来构建此类Schwartz形态。我们表明，构造扩散发生器的方法之一是使用常规拉格朗日。使用这种扩散生成器方法，我们还为SDE提供了扩展的ITO公式（也称为广义ITO公式或ITO-Wentzell的公式）。
A general way of representing Stochastic Differential Equations (SDEs) on smooth manifold is based on Schwartz morphism. In this manuscript we are interested in SDEs on a smooth manifold $M$ that are driven by p-dimensional Wiener process $W_t \in \mathbb{R}^p$. In terms of Schwartz morphism, such SDEs are represented by Schwartz morphism that morphs the semi-martingale $(t,W_t)\in\mathbb{R}^{p+1}$ into a semi-martingale on the manifold $M$. We show that it is possible to construct such Schwartz morphisms using special maps that we call as diffusion generators. We show that one of the ways of constructing diffusion generator is by using regular Lagrangian. Using this diffusion generator approach, we also give extended Ito formula (also known as generalized Ito formula or Ito-Wentzell's formula) for SDEs on manifold.
