General-order minimal recursion for conditional means and cumulants

Derive the minimal-recursion construction at arbitrary expansion order for the conditional mean and conditional cumulants in the finite-dimensional small-noise filtering framework.

Background

The paper establishes finite-dimensional closure for coefficient evolution and gives explicit low-order dimension counts for the conditional-mean expansion. It also shows that the same auxiliary variables suffice to compute the conditional characteristic function and the cumulants required at any fixed order.

However, the paper does not derive the minimal recursion at general expansion order. The unresolved task is to characterize and construct the smallest recursive finite-dimensional system governing both conditional-mean coefficients and cumulant corrections beyond the explicitly treated orders.