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

Abstract

We present a unified operator-theoretic framework for stochastic calculus based on the factorization (Id - E)F = {\delta}_X {\Pi}_X D_X F, valid for F_T^X-measurable F in L^2({\Omega}) when the driving process X has the representation property. For a square-integrable process X with stochastic integral {\delta}_X, we define the operator-covariant derivative D_X := {\delta}_X* as the Hilbert space adjoint of {\delta}_X. Combined with predictable projection {\Pi}_X, this yields a unified Clark-Ocone representation. The operator D_X F is defined as an adjoint for all F in L^2({\Omega}), 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.