On the Construction of the Neutrosophic Itˆo Integral

This research pioneers a rigorous framework for stochastic calculus in the neutrosophic paradigm, explicitly modeling systems with truth, indeterminacy, and falsehood degrees. We introduce foundational constructs including Canonical Neutrosophic Brownian Motion, Simple Neutrosophic Processes, and the space V2[0,T]. Building upon these, we formally define the Neutrosophic Itˆo Integral for ele mentary processes and extend it to general integrands in V2[0,T], establishing Neutrosophic Martingales, d-Dimensional Neutrosophic Brownian Motion, Matrix-Valued Neutrosophic Integrands, and the Multi-Dimensional Neutrosophic Itˆo Integral. The theory is further generalized through Local Neu trosophic Martingales, Locally Square-Integrable Neutrosophic Integrands, and Neutrosophic Itˆo Processes. Key theorems demonstrate the successful extension of classical stochastic calculus: The Density of Ele mentary Neutrosophic Processes enables integral construction, while the Neutrosophic Itˆo Isometry and its consistent extension ensure well-defined integration. Crucially, we prove Martingale Characteriza tion with Path Regularity, construct integrals for local martingales via stopping times, and establish the Itˆo Formula for both neutrosophic Brownian motion and general Itˆo processes. Significantly, essential structural properties–including isometry, martingale preservation, sample path conti nuity, localization efficacy, and the Itˆo formula–remain rigorously valid in this generalized uncertainty frame work. This work provides a powerful mathematical toolkit for stochastic systems under three-valued uncertainty, enabling applications across finance, engineering, and decision science.