Stochastic Calculus as Operator Factorization

Abstract: We present a unified operator-theoretic formulation of stochastic calculus based on two principles: fluctuations factor through differentiation, predictable projection, and integration, and the appropriate stochastic derivative is the Hilbert adjoint of the stochastic integral on the energy space of the driving process. On an isonormal Gaussian space we recover the identity (Id - E)F = delta Pi D F, where D is the Malliavin derivative, Pi is predictable projection, and delta is the divergence operator. Motivated by this factorization, we define for a square-integrable process X admitting a closed stochastic integral an operator-covariant derivative on L2(Omega) via Riesz representation. This yields a canonical Clark-Ocone representation that unifies Malliavin, Volterra-Malliavin, and functional Ito derivatives and clarifies the operator geometry underlying stochastic calculus.