Application of rough path theory for filtering and numerical integration methods

1998年T.里昂（Oxford）提出了一种新的方法来求解随机微分方程的鲁棒路径解，这种方法现在被称为粗糙路径分析。在此基础上，发展了一类新的求解随机微分方程的数值算法。最近，粗糙路径方法也被成功地推广到随机偏微分方程。在随机滤波中，带有加性噪声的马尔可夫信号的（未归一化）条件分布称为最优滤波器，它可以描述为随机偏微分方程（称为Zakai方程）的解。在建议的项目中，我们希望将粗糙路径分析应用于Zakai方程的鲁棒路径解，以构建鲁棒版本的最优滤波器。随后，我们要应用已知的算法的基础上的粗糙路径方法的数值逼近这些强大的估计，并进一步研究其性能。

In 1998 T. Lyons (Oxford) suggested a new approach for the robust pathwise solution of stochasticdifferential equations which is nowadays known as the rough path analysis. Based on thisapproach a new class of numerical algorithms for the solution of stochastic differential equationshave been developed. Recently, the rough path approach has been successfully extended also tostochastic partial differential equations. In stochastic filtering, the (unnormalized) conditionaldistribution of a Markovian signal observed with additive noise is called the optimal filter and itcan be described as the solution of a stochastic partial differential equation which is called theZakai equation. In the proposed project we want to apply the rough path analysis to a robustpathwise solution of the Zakai equation in order to construct robust versions of the optimal filter. Subsequently, we want to apply known algorithms based on the rough path approach to the numerical approximation of these robust estimators and further investigate their properties.