Blog
The Barrier Your Monte Carlo Never Sees
Discrete Monte Carlo monitoring silently under-counts barrier crossings with a stubborn sqrt(dt) bias — a one-line Brownian-bridge correction removes almost all of it.
Read more →When Sampling Faster Makes You Wrong
Realized variance should converge to an SDE's quadratic variation as you sample faster — market microstructure noise makes it diverge instead. The volatility signature plot, and the two-scale fix.
Read more →The Neuron as a Density: Fokker–Planck with a Reset
The same absorbing barrier, seen forward: the firing rate is a probability flux, the reset kinks the density, and a recurrent network has to solve for its own rate.
Read more →The Neuron as an Ornstein–Uhlenbeck Process
A leaky integrate-and-fire neuron's subthreshold voltage is an Ornstein-Uhlenbeck process, and its firing rate is a first-passage time — the same barrier-crossing math that prices a down-and-out option.
Read more →Cholesky vs Eigendecomposition: Two Ways to Sample Correlated Gaussians
Two matrix square roots for sampling correlated normals: Cholesky is several times faster, but the eigendecomposition survives a correlation matrix that isn't positive definite.
Read more →The 1/√N Wall: Why Monte Carlo Is Slow, and What Beats It
Monte Carlo error falls like 1/sqrt(N) no matter how clever your sampler. Quasi-Monte Carlo bends the exponent, not just the constant.
Read more →Euler–Maruyama vs Milstein: The Term That Matters
One extra term from the Itô–Taylor expansion doubles the strong order of an SDE solver — measured directly against an exact solution.
Read more →Integrating Oscillations: Filon vs Levin
Two classical methods for oscillatory integrals that beat the O(omega)-points curse — compared against an exact solution, with the log-error plot to prove it.
Read more →The Itô Integral: A Running Bet You Fix Before the Flip
Why the naive guess for the Itô integral is wrong, and the correction that makes stochastic calculus honest.
Read more →The Stagflation Trap: Optimal Monetary Policy as an HJB Problem
A linear-quadratic HJB model that makes the stagflation trap mathematically precise: why a persistent supply shock forces the optimal steady state away from zero inflation and zero output gap simultaneously.
Read more →De Rerum Stochástica: Lucretius’s Random Swerve and the Birth of SDEs
Epicurus added the clinamen — a random, uncaused swerve — to save atomism from determinism. Two thousand years later, it became the diffusion term in every SDE ever written.
Read more →From Plasma to Game Theory: The Unlikely Journey of an SDE
McKean-Vlasov SDEs emerged from plasma physics in the 1960s. Mean field games arrived from economics in 2006. They converged on the same equation from opposite directions.
Read more →Markov Chains in Supermarket Management
How Markov chains model checkout queues and customer flow — and what the stationary distribution tells a supermarket manager about staffing and store layout.
Read more →Kelly, the Growth-Optimal Portfolio, and the Benchmark Approach to Option Pricing
The Kelly criterion and Black-Scholes option pricing are the same mathematical object viewed from different angles. Both lead to the growth-optimal portfolio — Platen's benchmark — which prices derivatives under the real-world measure without requiring a risk-neutral one.
Read more →The Bookmaker’s Measure: Kelly, Martingale, and the Price of an Edge
The bookmaker is not running a gambling operation. They are constructing a risk-neutral measure. Understanding that changes everything about how you think about betting strategies — including why Martingale fails even when you have edge.
Read more →The Kelly Criterion — Why the Optimal Strategy Is Never Used
A 60/40 coin. A provably optimal betting strategy. And almost nobody uses it. Kelly's 1956 result is one of the few places in finance where the correct answer is known and rationally ignored.
Read more →Four Historical Decisions an ABM Would Have Changed
Smoot-Hawley was not unique. From Versailles to Vietnam to Stalin's grain quotas to the Maginot Line — the same pattern recurs. A model that forgets the other agents in the system.
Read more →The Trade War Nobody Chose: Smoot-Hawley and the Model That Would Have Stopped It
In 1930, a thousand economists begged Hoover not to sign the Smoot-Hawley tariff. He signed it anyway. An agent-based model would have shown exactly what happened next — before it happened.
Read more →CAD 530 a Month: What Your Car Insurance Premium Actually Pays For
A friend bought a Ford F-150 in Ontario. The insurance quote came in at 530 dollars a month. I built a mathematical model to find out how much of that is justified.
Read more →Langevin Dynamics and Why They Matter in Finance
The Langevin equation was written to describe a particle in a fluid. It turns out to be exactly the right language for interest rates, volatility, and forward curves.
Read more →Two Worlds, One Price: Entropy and the Risk-Neutral Measure
The change of measure at the heart of derivative pricing is an exponential tilt of the real-world probability measure — identical in structure to the Boltzmann distribution. The cost of that tilt is relative entropy.
Read more →Pricing a Gas Liquefaction Plant as a Real Option: An HJB Approach
We price the option to invest in a gas liquefaction plant using the Hamilton-Jacobi-Bellman PDE, with CIR gas price dynamics and a finite-horizon free boundary problem.
Read more →Heat Diffusion in an Iron Sphere — A FEniCSx Simulation
A 3D FEniCSx simulation of heat diffusing through an iron sphere with a hot core. Full PDE, boundary conditions, weak form, and DOLFINx code.
Read more →Solar Magnetohydrodynamics: Solving Magnetic Diffusion in a Sphere
Solving the MHD induction equation in spherical geometry with Dedalus — magnetic diffusion on the 2D sphere surface and inside a 3D conducting ball, with animated GIF simulations.
Read more →Solving the 2D Wave Equation with Spectral Methods
Using the Dedalus spectral PDE solver to simulate the 2D wave equation on a periodic domain, with animated GIF output and applications across physics, finance, and medical imaging.
Read more →The Square Root That Keeps Rain Positive: CIR Processes in Weather Modelling
The Cox-Ingersoll-Ross process was built for interest rates. Its real strength — staying positive as noise vanishes near zero — turns out to be exactly what rainfall modelling needs.
Read more →When Neurons Forget: The Ornstein-Uhlenbeck Process in Neuroscience
The leaky integrate-and-fire neuron — the workhorse model of computational neuroscience — is an Ornstein-Uhlenbeck process in disguise. Here is why that matters.
Read more →What Agent-Based Models See That Regressions Miss
Standard regression conflates micro configurations that produce identical macro observables. Four classical examples — wealth concentration from random exchange, Schelling segregation, phantom traffic jams, and bank runs — show that genuine causal structure can be invisible to aggregate statistics and only visible at the level of agent interactions.
Read more →The Mean Field Game of Mediocrity
Mean field game theory gives mediocracy a precise formulation: when conformity pressure exceeds rank incentive, the coupled HJB–Fokker–Planck system has a concentrated Nash equilibrium. A phase transition separates dispersed outcomes from universal indistinction.
Read more →Agent-Based Models in the Peanut Butter Industry
Any market with heterogeneous agents, inventory dynamics, and competitive pricing produces emergent behavior that ABM is built to capture. Peanut butter turns out to be a surprisingly clean example.
Read more →ABM or MFG? The Cases Where the Answer Is Not Clear
Agent-based models and mean field games are often presented as alternatives. In practice, a significant share of real problems sit between them — and recognizing that grey zone is the first step toward modeling it correctly.
Read more →