非线性滤波中小系统噪声展开的有限累积量塔
Finite Cumulant Towers for Small-System-Noise Expansions in Nonlinear Filtering
AI总结：
该研究针对小系统噪声下的非线性滤波，提出有限累积量塔方法，可递归求解条件累积量展开系数，减少常微分方程变量数，优于现有算法。
摘要：
我们研究小系统噪声 regime 下非线性滤波的渐近展开。主要结果是条件累积量的展开系数满足可递归求解的封闭有限维系统：在每个展开阶次，一个系数的演化会引入更高阶累积量的系数，而高阶消失特性终止该层级，形成有限三角塔。第一层是卡尔曼-布西滤波器，后续层在已知下层后满足线性方程。为推导这些方程，我们建立依赖观测的随机变量的广义扎凯方程和库什纳-斯特拉托诺维奇方程，得到任意维度下的显式系数方程。条件累积量展开也决定了条件特征函数的展开，我们进一步得到标量变量数关于状态维度和展开阶次的显式界。当观测函数消失时，该构造简化为无条件累积量系数的常微分方程。与现有渐近展开算法的比较表明，所得常微分方程变量数减少。
英文摘要：
We study asymptotic expansions for nonlinear filtering in the small-system-noise regime. Our main result is that the expansion coefficients of conditional cumulants satisfy a closed finite-dimensional system that can be solved recursively. At each expansion degree, the evolution of a coefficient introduces a coefficient of the next higher cumulant rank, while a high-rank vanishing property terminates this hierarchy, yielding a finite triangular tower. The first layer is the Kalman--Bucy filter, and subsequent layers satisfy linear equations once the lower layers are known. To derive these equations, we establish generalized Zakai and Kushner--Stratonovich equations for observation-dependent random variables and obtain explicit coefficient equations in arbitrary dimension. The conditional cumulant expansion also determines the expansion of the conditional characteristic function. We further obtain an explicit bound on the number of scalar variables in terms of the state dimension and expansion degree. When the observation function vanishes, the construction reduces to ODEs for unconditional cumulant coefficients. Comparisons with existing asymptotic-expansion algorithms illustrate the resulting reduction in the number of ODE variables.
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