An Operator Ito Formula for Volterra Gaussian Processes: The Intrinsic Bracket via Causal Derivation-Divergence Factorization

TL;DR

This paper develops an Ito-type change-of-variables formula for Volterra Gaussian processes, expressing the correction as a Stieltjes integral involving an energy function, and proves its uniqueness, invariance, and stability properties.

Contribution

It introduces an operator-based Ito formula for Volterra Gaussian processes, extending classical results and analyzing the intrinsic bracket and correction measures.

Findings

The energy measure d Gamma^X is the unique second-order correction.

The intrinsic bracket remains invariant under kernel representation changes.

The bracket is stable under L^2 kernel approximation.

Abstract

We derive an Ito-type change-of-variables formula for Volterra Gaussian processes (including fractional Brownian motion with any Hurst parameter), based on the operator factorization framework. The Ito correction is expressed as a Stieltjes integral against the energy function Gamma^X(t) := ||Pi DX_t||_H^2, which equals E[X_t^2] for centered Gaussian processes. The correction emerges from the non-commutativity of the predictable projection Pi with nonlinear functions and is computed via the Gaussian conditional expectation structure following Decreusefond-Ustunel. We prove three results beyond the formula itself: (1) the energy measure d Gamma^X is the unique second-order correction compatible with the operator factorization; (2) under a fixed driving martingale, the intrinsic bracket is invariant under changes of Volterra kernel representation; (3) the bracket is stable under L^2 kernel approximation. The proof uses a marginal density argument via the Gaussian heat equation, bypassing pathwise increments entirely. We restrict to first-order formulas; higher-order rough dynamics are not addressed. The proof relies on Gaussianity; for non-Gaussian processes the formula extends as a conjecture.