Dice Question Streamline Icon: https://streamlinehq.com

Joint Gaussianity of Increments in Chen–Hermite Almost Brownian Motion

Ascertain whether the disjoint increments of a Chen–Hermite almost Brownian motion are jointly Gaussian.

Information Square Streamline Icon: https://streamlinehq.com

Background

The classification of unbiased rough integrators narrows to processes with Brownian increments and Chen–Hermite balancing identities. However, the authors point out that beyond uncorrelated adjacent increments, it is not established whether the full joint Gaussian structure holds.

Determining joint Gaussianity of disjoint increments would bridge the gap between CHABM and standard Brownian motion, thereby completing the characterization of unbiased integrators in frictionless rough-path-based markets with rich strategy classes.

References

As remarked in Remark \ref{rem:CHABM vs BM}, its only difference with a Brownian motion is that we cannot tell whether the disjoint increments of a Chen-Hermite almost Brownian motion are jointly Gaussian.

— Unbiased Rough Integrators and No Free Lunch in Rough-Path-Based Market Models (2509.14529 - Ichiba et al., 18 Sep 2025) in Remark ‘CHABM vs BM’ in Subsection 4.2 (following the definition of Chen–Hermite almost Brownian motion)