Asymptotic Expansion of the Kallianpur-Striebel Formula under Perturbations of Linear Models

Abstract

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.