An Itô Formula via Predictable Projection for Non-Semimartingale Processes
Ramiro Fontes
TL;DR
This work presents an operator-centered Itô formula for processes that need not be semimartingales. By rewriting the fundamental identity as $(\mathrm{Id}-\mathbb{E}) = \delta \Pi D$ and replacing pathwise quadratic variation with the intrinsic bracket $\langle Y \rangle^{(D,\Pi)}$, the authors derive a change-of-variables formula that holds in $L^2$ for rough drivers such as Volterra Gaussian processes and fractional Brownian motion. The approach unifies semimartingale calculus, rough-path ideas, and Malliavin calculus under a single Hilbert-space framework, providing easy computation of the Itô correction via the energy $\int_0^t \|\Pi DY_s\|^2_H ds$.Key contributions include a rigorous product rule for divergences with sharp domain assumptions, an $L^2$-convergence proof of the Itô formula, multivariate extensions, a concrete treatment of Volterra Gaussian processes, and explicit worked examples that match classical results while avoiding complex rough-path constructions.
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.