Stochastic Calculus for Rough Fractional Brownian Motion via Operator Factorization

Abstract

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.