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