Chen–Hermite Almost Brownian Motion Is Brownian

Prove that any one-dimensional Chen–Hermite almost Brownian motion—namely, any stochastic process whose increments satisfy X_{s,t} ~ N(0, |t−s|) for all s,t and which additionally satisfies the Chen–Hermite balancing condition for every adjacent interval—must be a standard Brownian motion.

Background

The paper introduces the notion of a Chen–Hermite almost Brownian motion (CHABM) as a process with Brownian increment laws and additional ‘balancing’ identities annihilating mixed Hermite moments across adjacent increments. This concept arises in the classification of unbiased rough integrators when the admissible strategies include signature-type portfolios and early liquidation times.

While a standard Brownian motion satisfies the CHABM properties, the authors emphasize that it remains unclear whether CHABM assumptions force the full joint Gaussian increment structure, hence whether CHABM is exactly Brownian motion. Establishing this would finalize the classification by showing that the only unbiased integrator in rich strategy classes is (time-changed) Itô Brownian motion.