For a continuous process X with stochastic integral δX on a Hilbert energy space HX, the adjoint DX: = δX* defines an operator derivative via EF · δX (u) = E⟨DX F, u⟩₇ₗ. This paper develops a unified stochastic calculus for continuous processes from the factorization (Id − E) F = δX (ΠX DX F), structured into two layers: a representation layer (adjoint alone) and a calculus layer (Leibniz rule). Representation layer. The factorization yields a unified Clark–Ocone formula when X has the predictable representation property, and the optimal L²-approximation (Galtchouk–Kunita–Watanabe projection) when it does not. For Gaussian Volterra processes, ΠX DX F coincides with the Gubinelli derivative of rough path theory. Calculus layer. The Leibniz rule DX (FG) = F DX G + G DX F is logically independent of the adjoint definition. We prove it holds whenever DX admits a cylindrical reduction—an integration-by-parts formula reducing DX to partial differentiation on a dense class. In the Hilbert space setting, the Riesz identification conceals the distinction between duality and adjointness, allowing the factorization to work seamlessly for all continuous processes admitting such a reduction. When Leibniz holds, the factorization yields an Itô formula with intrinsic bracket ∫₀ᵗ ‖ΠX DX Yₛ‖²₇ₗ ds serving as the analog of pathwise quadratic variation. Stochastic volatility. For non-Gaussian continuous martingales Mₜ = ∫₀ᵗ σₛ dWₛ, we prove that deterministic cylindrical reduction is impossible (the obstruction), but Leibniz holds via random cylindrical reduction with representers κᵢ (t) = DₜW (Mₓ㶁) /σₜ. This mechanism depends on the underlying Brownian driver—necessarily so, since the obstruction proves no intrinsic deterministic structure can work. This identifies the boundary of the method: filtrations not generated by Brownian motion, where the random cylindrical reduction is unavailable. Scope and sequel. The theory developed here is restricted to continuous processes on Hilbert energy spaces. For jump processes (Poisson, Lévy), the one-parameter operator derivative DX cannot accommodate the nonlinear integrands arising from finite-difference operators; centered functionals lie in the kernel of the derivative, revealing a representability obstruction that requires a two-parameter formulation on Banach energy spaces. These structural limitations and their resolution are developed in a companion paper (doi: 10. 5281/zenodo. 18625711).
Ramiro Fontes (Thu,) studied this question.