The Operator Derivative in Stochastic Calculus

This work develops a unified operator-theoretic framework for stochastic calculus in separable Hilbert spaces, extending classical Itô and Clark–Ocone theory through a structurally explicit derivative–divergence factorization. We construct a densely defined, closable stochastic derivative operator and its adjoint, establishing a projection-based representation of stochastic integrals via predictable subspaces of L2 (Ω;H) L² (; H) L2 (Ω;H). The resulting formulation isolates the geometric structure underlying stochastic integration and clarifies the relationship between Malliavin-type operators and Itô representations. The framework yields: A representation theorem for square-integrable functionals via orthogonal projection onto predictable processes. A structural interpretation of the Clark–Ocone formula as an operator factorization. Extensions applicable to fractional and rough regimes (including Hurst H<1/2H < 1/2H<1/2). A formulation that naturally generalizes toward Banach-space settings. Rather than introducing a new calculus, the work reorganizes existing stochastic integration theory into a coherent operator structure that exposes its underlying geometry. This perspective clarifies domain issues, adjoint structure, and predictable projections in a unified setting. The results are intended to serve both probabilists and researchers working at the interface of stochastic analysis, SPDE theory, and operator methods.