Stochastic Calculus as Operator Factorization An Operator-Covariant Derivative and Unified Representation

We present a unified operator-theoretic framework for stochastic calculus based on the factorization (Id - E)F = δ_X Π_X D_X F, valid for F_T^X-measurable F in L^2(Ω) when the driving process X has the representation property. For a square-integrable process X with stochastic integral δ_X, we define the operator-covariant derivative D_X := δ_X* as the Hilbert space adjoint of δ_X. Combined with predictable projection Π_X, this yields a unified Clark-Ocone representation. The operator D_X F is defined as an adjoint for all F in L^2(Ω), without differentiability assumptions; the representation holds when X has the predictable representation property, and reduces to the Galtchouk-Kunita-Watanabe projection when it does not. The framework requires no reproducing kernel Hilbert space or Cameron-Martin structure, and applies to non-Gaussian processes. We work out concrete examples including Brownian motion, general continuous martingales, and compensated Poisson processes.