AI 中文总结
本文提出一种有限维递推方法，用于高效计算非线性滤波中小噪声渐近展开的系数，将计算复杂度从指数降为多项式，并推广至条件特征函数。
AI 中文摘要
本文针对非线性滤波中的小系统噪声渐近展开，提供了计算其系数的递推公式。该展开基于Kallianpur--Striebel公式获得，并在作者先前的工作中得到了证明。我们的主要贡献在于，通过应用Fubini定理和Wick公式并对所得项进行微分，将系数计算简化为一个扩展Kalman--Bucy滤波器的有限维系统。对于每个固定的展开阶数，变量数目随系统维度最多以多项式速度增长，而非指数增长。为证明构造的合理性，我们将所需的非适应积分定义为离散和的极限，并建立了一个广义Ito公式。我们还将展开从条件期望推广到条件特征函数，并提供了该方法的数值示例。
英文摘要
This paper provides a recursive formula for computing the coefficients in a small-system-noise asymptotic expansion for nonlinear filtering. The expansion, obtained from the Kallianpur--Striebel formula, was justified in the author's previous work. Our main contribution is to reduce the coefficient calculation to a finite-dimensional system extending the Kalman--Bucy filter by applying Fubini's theorem and Wick's formula and differentiating the resulting terms. For each fixed expansion order, the number of variables grows at most polynomially, rather than exponentially, with the system dimension. To justify the construction, we define the required non-adapted integrals as limits of discrete sums and establish a generalized Ito formula. We also extend the expansion from conditional expectations to conditional characteristic functions and provide a numerical illustration of the method.
