Tag
Stochastic Processes
My approach to stochastic modeling is grounded in non-stationary SDE theory: the view that financial price dynamics are better described by a changing diffusion manifold than by a fixed parametric model. The Polybius engine models price dynamics as probability mass flow under a modified Fokker-Planck operator, using a Nyström diffusion map to track the evolving market manifold and random matrix theory to classify execution regimes in real time.
Blog
April 8, 2026
Probability and Statistics: A Geometric Foundation
A measure-theoretic construction of probability and statistics, from sigma-algebras through estimation theory and hypothesis testing to the Riemannian geometry of statistical manifolds.
March 27, 2026
The Kelly Criterion from Shannon Information Theory
A rigorous derivation of Kelly's growth-rate-optimal betting strategy from Shannon's mutual information, with application to binary prediction markets.
March 17, 2026
Price as Geometry: Resolution, Coarse-Graining, and the Structure of Market Noise
A rigorous tour through stationary and non-stationary models of price evolution, with geometric analysis at the forefront. From the random walk null and Black-Scholes as flat geometry, through mean reversion as curved Riemannian diffusion, wavelets, geometric harmonics, and information geometry, anchored throughout by empirical evidence from BTC/ETH millisecond data.
March 10, 2026
Manifolds: The Language of Modern Geometry
A rigorous construction of smooth manifolds from first principles: charts, tangent spaces, Riemannian metrics, curvature tensors, and geometric flows.
March 10, 2026
Kuramoto: How Order Emerges from Chaos
Fireflies, neurons, power grids: all governed by the same equation. A tour through the Kuramoto model, its order parameter, and the phase transition that turns noise into rhythm.
Engineering
Pathwise
High-performance SDE simulation toolkit. Rust core with Python bindings via PyO3. Supports Brownian motion, GBM, Ornstein-Uhlenbeck, and custom processes with rayon-parallelised Monte Carlo.
Elworthy
JIT compiler that specialises Bismut-Elworthy-Li formulas into SIMD kernels for unbiased Monte Carlo Greeks on non-stationary SDEs. Symbolic AST, Cranelift lowering (scalar and 2-lane F64X2), multi-dimensional Heston driver, pathwise and likelihood-ratio Malliavin parameter Greeks (machine-checked with SymPy). European call price and BEL delta cross-validated against Black-Scholes closed form and the independent blackscholes crate; both agree within four Monte Carlo standard errors. About 22x over a tree-walking interpreter on GBM paths.
Kloeden
Hand-written SIMD C++ vs Rust (LLVM + Cranelift) benchmark companion to pathwise and elworthy. Same Brownian-increment fixture across four impls; single-thread pinned-core throughput on scalar Euler / Milstein / Taylor 1.5 on GBM, plus a digital-delta correctness table showing naive pathwise silently returns 0 in both languages while the Bismut-Elworthy-Li constant-flow weight matches analytic within 4 Monte Carlo standard errors (bitwise-identical between hand-rolled C++, hand-rolled Rust, and elworthy_rt::from_paths). Named after Peter Kloeden.