Stochastic Calculus for Rough Fractional Brownian Motion via Operator Factorization
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910004817166336 |
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| author | Fontes, Ramiro |
| author_facet | Fontes, Ramiro |
| contents | We develop an operator-theoretic formulation of stochastic calculus for fractional Brownian motion with Hurst parameter H in (0, 1/2). The approach is based on adjointness between stochastic integration and differentiation in the Cameron-Martin space of the driving process. For Gaussian Volterra processes, we establish a canonical factorization of fluctuations (Id - E) = delta_X Pi_X D_X, where D_X := delta_X^* is the operator-covariant derivative (adjoint of the stochastic integral), delta_X the divergence, and Pi_X the predictable projection. In the rough fractional regime, the factorization yields explicit derivative formulas for cylindrical functionals, controlled expansions of conditional expectations with O(|t-s|^{2H}) remainders, and an intrinsic identification of the Gubinelli derivative as the predictable component Pi_X D_X F. The framework extends to mixed semimartingale-rough processes, providing a unified calculus without requiring iterated integrals or signature constructions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09967 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stochastic Calculus for Rough Fractional Brownian Motion via Operator Factorization Fontes, Ramiro Probability Primary 60H07, Secondary 60G15, 47N30 We develop an operator-theoretic formulation of stochastic calculus for fractional Brownian motion with Hurst parameter H in (0, 1/2). The approach is based on adjointness between stochastic integration and differentiation in the Cameron-Martin space of the driving process. For Gaussian Volterra processes, we establish a canonical factorization of fluctuations (Id - E) = delta_X Pi_X D_X, where D_X := delta_X^* is the operator-covariant derivative (adjoint of the stochastic integral), delta_X the divergence, and Pi_X the predictable projection. In the rough fractional regime, the factorization yields explicit derivative formulas for cylindrical functionals, controlled expansions of conditional expectations with O(|t-s|^{2H}) remainders, and an intrinsic identification of the Gubinelli derivative as the predictable component Pi_X D_X F. The framework extends to mixed semimartingale-rough processes, providing a unified calculus without requiring iterated integrals or signature constructions. |
| title | Stochastic Calculus for Rough Fractional Brownian Motion via Operator Factorization |
| topic | Probability Primary 60H07, Secondary 60G15, 47N30 |
| url | https://arxiv.org/abs/2601.09967 |