Application of rough path theory for filtering and numerical integration methods

粗糙路径理论在滤波和数值积分方法中的应用

• 批准号: 203014755
• 负责人: Professor Dr. Peter Karl Friz
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2011
• 资助国家: 德国
• 起止时间: 2010-12-31 至 2015-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/203014755?language=en
• 关键词: Application; rough; path; theory; filtering

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.

1998年，T. Lyons（牛津）提出了随机微分方程鲁棒路径解的新方法，即现在所知的粗糙路径分析。在此基础上，提出了求解随机微分方程的一类新的数值算法。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，观察到的带有加性噪声的马尔可夫信号的（非归一化）条件分布称为最优滤波器，它可以被描述为随机偏微分方程的解，称为zakai方程。在本项目中，我们希望将粗糙路径分析应用于Zakai方程的鲁棒路径解，以构建最优滤波器的鲁棒版本。随后，我们希望将基于粗糙路径方法的已知算法应用于这些鲁棒估计量的数值逼近，并进一步研究它们的性质。