What question did this study set out to answer?

The aim is to extend operator factorization from Hilbert to Banach spaces and explore implications for Lévy processes.

February 14, 2026Open Access

Operator Factorization Beyond Hilbert Spaces: Banach Duality and Stable Levy Processes

Key Points

The aim is to extend operator factorization from Hilbert to Banach spaces and explore implications for Lévy processes.
Extended stochastic calculus framework to Banach energy spaces.
Defined the operator derivative D_X as the Banach dual of the stochastic integral.
Derived a product rule with Leibniz defect for symmetric γ-stable Lévy processes.
Established a fluctuation factorization under certain conditions.
Identified the non-Hilbert nature of the integrand space L^p for p < γ < 2.
Measured Leibniz defect without relying on quadratic variation.

Abstract

We extend the operator factorization framework for stochastic calculus from Hilbert to Banach energy spaces. The operator derivative DX, defined as the Banach dual of the stochastic integral, yields a fluctuation factorization under an explicit representation hypothesis (Theorem A). The principal result (Theorem B) is a product rule with Leibniz defect for symmetric γ-stable Lévy processes (γ ∈ (1, 2) ): the integrand space Lᵖ with p < γ < 2 is non-Hilbert, and the failure of the Leibniz rule is measured by the sum of product jump defects — without reference to quadratic variation, which is infinite in this setting.

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