论文标题
曲面上的曲线方案
Curved Schemes for SDEs on Manifolds
论文作者
论文摘要
给定$ \ mathbb {r}^n $中的随机微分方程（SDE），其解决方案被限制在某些歧管$ M \ subset \ mathbb {r}^n $中，我们为SDE提供了一类数字方案，该迭代率接近$ m $。我们的方案在几何上不变，可以选择为任何具有差异化至$ n $二维的布朗尼运动的SDE提供完美的解决方案。与基于投影的方法不同，我们的方案可以在没有明确知识的情况下实施M。我们的方法不需要模拟超出实施Euler-Maryuama计划所需的任何迭代的ItôEnterals。我们证明该方案在一组标准的假设下汇合，并通过考虑开普勒问题的随机版本来说明它们的实际优势。
Given a stochastic differential equation (SDE) in $\mathbb{R}^n$ whose solution is constrained to lie in some manifold $M \subset \mathbb{R}^n$, we propose a class of numerical schemes for the SDE whose iterates remain close to $M$ to high order. Our schemes are geometrically invariant, and can be chosen to give perfect solutions for any SDE which is diffeomorphic to $n$-dimensional Brownian motion. Unlike projection-based methods, our schemes may be implemented without explicit knowledge of M. Our approach does not require simulating any iterated Itô interals beyond those needed to implement the Euler--Maryuama scheme. We prove that the schemes converge under a standard set of assumptions, and illustrate their practical advantages by considering a stochastic version of the Kepler problem.
