🤖 AI Summary
This work addresses the absence of rigorous formalization of Itô integration and Itô’s formula in existing proof assistants, particularly the lack of machine-verified treatment of these constructs as martingale processes. Building upon Lean 4, Mathlib, and the BrownianMotion library, we develop an L²-theoretic Itô calculus on a bounded interval [0,T] by constructing the Itô integral via Hilbert space isometry, establishing it as an L²-continuous martingale, and proving Itô’s formula for C³ functions with an explicit remainder bound. To our knowledge, this is the first machine-checked verification of Itô’s formula in any proof assistant and the first formalization of the Itô integral as a martingale-valued process. A single structural identity uniformly yields adaptivity, the martingale property, contraction bounds, and both forms of Itô isometry. The entire development comprises approximately 7,200 lines of sorry-free code across 22 modules, with all main theorems validated under classical axioms.
📝 Abstract
We present a machine-checked development of the $L^2$ Itô calculus of Brownian motion on a bounded time interval $[0,T]$, formalized in Lean 4 on top of Mathlib and the BrownianMotion package. The development contains: the construction of the Itô integral as an isometry of Hilbert spaces, from a predictable-rectangle $π$-system through the density of simple adapted processes; the Itô integral as a process, proved to be an $L^2$-continuous martingale through a single structural identity (the integral at time $t$ is the conditional-expectation projection of its terminal value onto $\mathcal{F}t$), from which adaptedness, the martingale property, the contraction bound, and both the terminal and the time-indexed Itô isometries follow as corollaries; and Itô's formula for $C^3$ functions with bounded derivatives, including its time-dependent form $df = f_x,dB + (f_t + \tfrac12 f{xx}),dt$, obtained by a discrete-to-continuous argument through weighted quadratic variation and explicit $L^2$ remainder bounds. To our knowledge this includes the first machine-checked proof of Itô's formula, and the first machine-checked construction of the Itô integral as a martingale-valued process, in any proof assistant. We are deliberate about the boundary: the theory is the $L^2$ theory on $[0,T]$ with bounded-derivative integrand classes; localization to the unrestricted $C^2$ formula, integrators beyond Brownian motion, and pathwise statements are out of scope, and we say precisely why and where. The development is roughly 7,200 lines of Lean across 22 modules; every theorem is sorry-free, the axioms of each headline result are pinned to Mathlib's classical defaults by a build-enforced gate, and the whole is reproducible from a pinned toolchain.