An Operator Ito Formula for Volterra Gaussian Processes: The Intrinsic Bracket via Causal Derivation-Divergence Factorization
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908837128175616 |
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| author | Fontes, Ramiro |
| author_facet | Fontes, Ramiro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10359 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Operator Ito Formula for Volterra Gaussian Processes: The Intrinsic Bracket via Causal Derivation-Divergence Factorization Fontes, Ramiro Probability 60H07, 60H05, 60G15, 60G22 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. |
| title | An Operator Ito Formula for Volterra Gaussian Processes: The Intrinsic Bracket via Causal Derivation-Divergence Factorization |
| topic | Probability 60H07, 60H05, 60G15, 60G22 |
| url | https://arxiv.org/abs/2601.10359 |