Systematic cumulant reorganization of the finite polynomial correction

Develop a systematic reorganization of the finite polynomial correction to the conditional characteristic-function expansion in terms of conditional cumulants, including the necessary control of cumulant orders and the resulting exponential remainder.

Background

The paper proves that, at every fixed order in the small-noise parameter, the conditional characteristic function can be represented as the characteristic function of the leading Gaussian approximation multiplied by a finite polynomial. The authors note that this polynomial representation can alternatively be expressed using conditional cumulants.

Such a reformulation is not established because it requires additional estimates controlling the relevant cumulant orders and the remainder generated when the cumulant expansion is exponentiated. The problem is therefore to provide a rigorous systematic cumulant-based organization of the correction terms.

References

The finite polynomial correction can equivalently be reorganized in terms of conditional cumulants. Since such a reorganization requires additional control of cumulant orders and of the resulting exponential remainder, we leave its systematic treatment to subsequent work.

— Finite-Dimensional Recursions for Small-Noise Expansions in Nonlinear Filtering  (2609.18229 - Kurisaki, 16 Sep 2026) in Remark following Proposition 5.1, Section 5, “Expansion of the conditional distribution”

Deriving an expansion of the conditional density of $X_t\epsilon$ by Fourier inversion requires remainder estimates that justify integration over $u$, beyond estimates for each fixed $u$. Establishing such estimates is beyond the scope of this paper and is left for future work.

— Finite Cumulant Towers for Small-System-Noise Expansions in Nonlinear Filtering  (2610.21986 - Kurisaki, 15 Oct 2026) in Remark following Proposition 5.?, subsection “Expansion of the conditional characteristic function”