arXiv 1709.021152017-09-16math.PR

随机微分方程流的光滑性与逐路唯一性

Smoothness of Flow and Path-by-Path Uniqueness in Stochastic Differential Equations

Siva Athreya, Suprio Bhar, Atul Shekhar

备注：: Comments are welcome

摘要：

考虑随机微分方程$$ X_t = x_0 + \int_0^t f(X_s)ds + \int_0^tσ(X_s)dB^{H}_s,$$（$x_0 \in \mathbb{R}^d$、$d \geq 1$、$f: \mathbb{R}^d \rightarrow \mathbb{R}^d$）是有界连续的，$σ: \mathbb{R}^d \rightarrow \mathbb{R}^{d\times d}$是一致椭圆、有界、二次连续可微的保守向量场，$B^H$是分数布朗运动，具有$H \in (\frac{1}{3}, \frac{1}{2}]$。当$d=1$、$H= \frac{1}{2}$和$f$是Hölder连续的时，根据Davie [D 07]的精神，我们建立了一个只依赖于$f, σ$的空集$\mathcal{N}$的存在性，使得对于所有的$x_0\in \mathbb{R}$和$ω\in Ω\setminus \mathcal{N}$，上面的方程允许一个逐路唯一解。我们的证明是建立在与方程相关的流的一致连续可微性的基础上的。我们还建立了$d \geq 1$和$H \in (\frac{1}{3}, \frac{1}{2}]$的逐路唯一性，但空集可能依赖于$x_0$，从而推广了Catellier-Gubinelli [CG 12]的结果。

英文摘要：

We consider the stochastic differential equation $$ X_t = x_0 + \int_0^t f(X_s)ds + \int_0^tσ(X_s)dB^{H}_s,$$ with $x_0 \in \mathbb{R}^d$, $d \geq 1$, $f: \mathbb{R}^d \rightarrow \mathbb{R}^d$ is bounded continuous, $σ: \mathbb{R}^d \rightarrow \mathbb{R}^{d\times d}$ is a uniformly elliptic, bounded, twice continuously differentiable conservative vector field and $B^H$ is fractional Brownian motion with $H \in (\frac{1}{3}, \frac{1}{2}]$. When $d=1$, $H= \frac{1}{2}$, and $f$ is Hölder continuous, in the spirit of Davie [D07], we establish the existence of a null set $\mathcal{N}$ depending only on $f, σ$ such that for all $x_0\in \mathbb{R}$ and $ω\in Ω\setminus \mathcal{N}$, the above equation admits a path-by-path unique solution. Our proof is based on establishing the uniform continuous differentiability of the flow associated with the equation. We also establish the path-by-path uniqueness for $d \geq 1$ and $H \in (\frac{1}{3}, \frac{1}{2}]$, but the null set may depend on $x_0$, thus extending a result of Catellier-Gubinelli [CG12].

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文