线性模型扰动下Kallianpur-Striebel公式的渐近展开
Asymptotic Expansion of the Kallianpur-Striebel Formula under Perturbations of Linear Models
中文总结 AI 辅助
本文研究作为线性模型扰动的非线性状态空间模型，论证了Kallianpur-Striebel公式渐近展开的理论基础，其系数可通过常微分方程组计算，为非线性滤波提供新方法。
中文摘要 AI 辅助
本文研究一类作为线性模型扰动的非线性状态空间模型，针对该非线性滤波器关于扰动参数的渐近展开展开研究。给定观测值的隐状态的条件期望可通过Kallianpur-Striebel公式得到闭式表达；我们在非常一般的假设下，严格论证了该表达式在概率意义下的展开合理性。我们的框架不仅适用于近线性模型，还适用于更一般的情形，包括带有小系统噪声的非线性系统。在这些情形下，所得展开式的系数可通过常微分方程组计算；尽管具体计算流程将在后续论文中给出，但本工作为非线性滤波的渐近展开方法奠定了理论基础。
英文摘要
In this paper, we study a nonlinear state-space model represented as a perturbation of a linear model and investigate an asymptotic expansion of the nonlinear filter with respect to the perturbation parameter. The conditional expectation of the hidden state given the observations admits a closed-form representation via the Kallianpur-Striebel formula. We provide a rigorous justification of the expansion of this representation in probability under very general assumptions. Our framework applies not only to nearly linear models but also to more general situations, including nonlinear systems with small system noise. In these cases, the coefficients of the resulting expansion can be computed through systems of ordinary differential equations. Although the explicit computational procedure will be presented in a subsequent paper, the present work establishes the theoretical foundation of our asymptotic expansion approach to nonlinear filtering.
