A Machine-Checked Itô Calculus for Brownian Motion

Battle Count: 78

This paper is foundational rather than directly applicable to trading strategies. It provides the first machine-checked Itô calculus, which is the mathematical backbone of continuous-time quantitative finance (Black-Scholes, stochastic volatility models, interest rate models). The parent library uses this to derive pricing formulas rather than assume them. However, the paper itself does not address trading signals, portfolio construction, execution, or market microstructure. Its relevance is at the level of formal correctness assurance for the mathematical infrastructure underlying quantitative finance.

Extremely high complexity: ~7,900 lines of Lean 4 across 26 modules, ~250 theorems/lemmas, ~40 definitions. Requires deep expertise in both stochastic calculus (measure-theoretic probability, Lp spaces, conditional expectation, Gaussian measures, martingale theory) and Lean 4/Mathlib formalization (continuous linear maps, dense-range extensions, trimmed measures, π-λ theorem, dominated convergence in formal settings). The proof architecture involves multiple interacting layers: discrete-to-continuous limits, weighted quadratic variation, Riemann bridges, conditional-expectation projections, maximal inequalities, Borel-Cantelli arguments, and gluing constructions across horizons. Not a paper one can 'implement' in a conventional sense; it is a complete formal mathematical development.

Exceptionally reproducible: pinned toolchain, pinned upstream dependency (BrownianMotion at commit eaa4391), no sorry or admit anywhere, build-enforced axiom audit pinning all headline results to Mathlib's classical defaults (propext, Classical.choice, Quot.sound), CI gate fails build on new axioms, hash ledger records exact source inputs per verified statement, continuous-integration gate, and full artifact available on GitHub. Every displayed theorem maps to a specific Lean declaration name and module (Appendix A).