[论文解读] Expected signature of two dimensional Brownian Motion up to the first exit time of the domain

Terry Lyons, Hao Ni|arXiv (Cornell University)|Jan 31, 2011

Markov Chains and Monte Carlo Methods参考文献 1被引用 5

一句话总结

本文研究有界区域边界吸收的二维布朗运动的期望签名，表明其满足带有边界条件的椭圆PDE系统。在光滑性和有界性假设下，递归Sobolev估计可得出签名项的几何衰减界，尽管即使在单位圆盘情形下，签名分布律的完全确定性仍未得到证明。

ABSTRACT

The signature of a path provides a top down description of the path in terms of its effects as a control [Differential Equations Driven by Rough Paths (2007) Springer]. The signature transforms a path into a group-like element in the tensor algebra and is an essential object in rough path theory. The expected signature of a stochastic process plays a similar role to that played by the characteristic function of a random variable. In [Chevyrev (2013)], it is proved that under certain boundedness conditions, the expected value of a random signature already determines the law of this random signature. It becomes of great interest to be able to compute examples of expected signatures and obtain the upper bounds for the decay rates of expected signatures. For instance, the computation for Brownian motion on $[0,1]$ leads to the ``cubature on Wiener space'' methodology [Lyons and Victoir, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004) 169-198]. In this paper we fix a bounded domain $\Gamma$ in a Euclidean space $E$ and study the expected signature of a Brownian path starting at $z\in\Gamma$ and stopped at the first exit time from $\Gamma$. We denote this tensor series valued function by $\Phi_{\Gamma}(z)$ and focus on the case $E=\mathbb{R}^d$. We show that $\Phi_{\Gamma}(z)$ satisfies an elliptic PDE system and a boundary condition. The equations determining $\Phi_{\Gamma}$ can be recursively solved; by an iterative application of Sobolev estimates we are able, under certain smoothness and boundedness condition of the domain $\Gamma$, to prove geometric bounds for the terms in $\Phi_{\Gamma}(z)$. However, there is still a gap and we have not shown that $\Phi_{\Gamma}(z)$ determines the law of the signature of this stopped Brownian motion even if $\Gamma$ is a unit ball.

研究动机与目标

• 表征二维布朗运动在有界区域首次首出时间停止时的期望签名。
• 建立 governing 期望签名张量级数的椭圆PDE系统。
• 在区域光滑性和有界性条件下，推导期望签名中各项的几何衰减界。
• 研究期望签名在多大程度上能确定停止布朗运动签名的分布律。

提出的方法

• 将期望签名建模为定义在区域Γ上的张量级数取值函数ΦΓ(z)。
• 基于布朗运动的生成元和伊藤公式，推导ΦΓ(z)所满足的椭圆PDE系统。
• 递归应用Sobolev估计以控制签名展开中各项张量分量的增长。
• 利用区域的光滑性和有界性假设，确保签名项衰减率的统一控制。
• 利用张量代数中签名的群性质，分析随机积分下系统的行为。
• 分析由首次首出时间停止规则引出的边界条件，将其与PDE公式联系起来。

实验结果

研究问题

• RQ1 二维布朗运动在首次首出时间的期望签名是否满足具有边界条件的明确定义的椭圆PDE系统？
• RQ2 在光滑且有界区域假设下，能否为期望签名中的各项建立几何衰减界？
• RQ3 期望签名在多大程度上能确定停止布朗运动签名的分布律？
• RQ4 即使在单位圆盘等简单区域中，期望签名是否足以重构停止路径签名的完整分布？
• RQ5 Sobolev估计在控制高阶签名分量增长方面起到何种作用？

主要发现

• 期望签名ΦΓ(z)满足由首次首出时间停止规则导出的具有边界条件的椭圆PDE系统。
• 在区域Γ的光滑性和有界性条件下，递归Sobolev估计可得出ΦΓ(z)张量级数展开中各项的几何衰减界。
• 签名分量的衰减率被证明为几何级数，意味着张量项阶数的指数衰减。
• 尽管有这些界，本文仍未证明ΦΓ(z)唯一确定停止布朗运动签名的分布律。
• 该方法未能填补证明期望签名可完全确定分布律的缺口，即使在单位球区域中亦然。