Gaussianity from Chen–Hermite Balancing and Gaussian Single Increments

Show that any one-dimensional continuous stochastic process whose single increments are Gaussian and which satisfies the Chen–Hermite balancing condition for all adjacent intervals has multivariate normal finite-dimensional distributions; equivalently, establish that such a process is Gaussian.

Background

The authors discuss that CHABM has the same variance and covariance structure as Brownian motion, and that level-2 balancing yields uncorrelated adjacent increments. They note that proving joint Gaussianity of increments would imply that CHABM is Brownian.

This conjecture targets a more general implication: from Gaussian single increments plus Chen–Hermite balancing across adjacent intervals, deduce that all finite-dimensional distributions are jointly Gaussian. Resolving this would substantiate the stronger CHABM-is-Brownian conjecture.