论文标题
最小曲率流量和Martingale退出时间
Minimum curvature flow and martingale exit times
论文作者
论文摘要
我们研究以下问题：适当地归一化的martingale $ x $可以将最大的确定性时间$ t _*$保存在cosex of $ \ mathbb {r}^d $中？我们在粘度框架中表明，$ t _*$等于$ k $所需的时间以达到$ x（0）$，因为它经历了我们称为（正）最小曲率流的几何流量。该结果与几何流动的随机和游戏表示的文献有着密切的联系。此外，最小曲率流可以看作是Ambrosio-soner codimension-$（d-1）$平均曲率流的到达时间版本，$ 1 $ skeleton的$ k $。我们的结果是通过概率和分析方法的混合而获得的。
We study the following question: What is the largest deterministic amount of time $T_*$ that a suitably normalized martingale $X$ can be kept inside a convex body $K$ in $\mathbb{R}^d$? We show, in a viscosity framework, that $T_*$ equals the time it takes for the relative boundary of $K$ to reach $X(0)$ as it undergoes a geometric flow that we call (positive) minimum curvature flow. This result has close links to the literature on stochastic and game representations of geometric flows. Moreover, the minimum curvature flow can be viewed as an arrival time version of the Ambrosio--Soner codimension-$(d-1)$ mean curvature flow of the $1$-skeleton of $K$. Our results are obtained by a mix of probabilistic and analytic methods.
