Identifiability of temporal weights within the fitting network family

Determine whether nonuniqueness of smooth, strictly positive, unit-mean temporal weights that induce the same increment covariance matrix occurs within the particular neural-network family used to fit the temporal weight in weighted sub-fractional Brownian motion Euler models.

Background

The Euler likelihood on a fixed observation grid depends on the temporal weight only through the finitely many entries of the driving-increment covariance matrix and the imposed unit-mean normalization. The appendix constructs a nonzero smooth perturbation of the constant weight whose integral is zero and whose contribution to every covariance-matrix entry vanishes. Consequently, distinct smooth positive weights with unit mean can generate exactly the same Euler observation law when the drift, diffusion, and initial state are fixed.

This construction establishes nonuniqueness in the broad class of smooth positive weights, but it does not resolve whether the same phenomenon occurs after restricting the weight to the specific neural-network representation used for estimation. Establishing or refuting identifiability in that restricted representation would clarify whether the fitted temporal-weight parameters can be uniquely determined from observations on a fixed grid.