An Itô Formula via Predictable Projection for Non-Semimartingale Processes

Abstract: We derive an Itô-type change-of-variables formula for a class of adapted stochastic processes that do not necessarily admit semimartingale structure. The formulation is based on an intrinsic Hilbert-space derivative together with a predictable projection operator, allowing stochastic integrals to be expressed without reliance on quadratic variation or anticipative calculus. The resulting formula replaces the classical quadratic variation term with a computable second-order contribution expressed as a norm of the projected derivative. In the semimartingale case, the formula reduces to the classical Itô formula. The approach applies naturally to processes with memory and non-Markovian dependence, providing a unified and intrinsic framework for stochastic calculus beyond the semimartingale setting.