The Operator Derivative in Continuous Stochastic Calculus: A Hilbert Energy Space Framework

For a continuous process X with stochastic integral δ_X on a Hilbert energy space H_X, the adjoint D_X := δ_X* defines an operator derivative via E[F · δ_X(u)] = E[⟨D_X F, u⟩H_X]. This paper develops a unified stochastic calculus for continuous processes from the factorization (Id − E)F = δ_X(Π_X D_X 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, Π_X D_X F coincides with the Gubinelli derivative of rough path theory. Calculus layer. The Leibniz rule D_X(FG) = F D_X G + G D_X F is logically independent of the adjoint definition. We prove it holds whenever D_X admits a cylindrical reduction—an integration-by-parts formula reducing D_X 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 ‖Π_X D_X Y_s‖²H_X ds serving as the analog of pathwise quadratic variation. Stochastic volatility. For non-Gaussian continuous martingales M_t = ∫₀ᵗ σ_s dW_s, we prove that deterministic cylindrical reduction is impossible (the obstruction), but Leibniz holds via random cylindrical reduction with representers κ_i(t) = D_t^W(Mt_i)/σ_t. 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 D_X 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).