Finite-Dimensional Recursions for Small-Noise Expansions in Nonlinear Filtering

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.