De Rerum Stochástica: Lucretius’s Random Swerve and the Birth of SDEs I. The Void and Its Grains Leucippus of Miletus, near 450 BC, first spoke the word — atoms and the void, the only truth he heard. Before him: gods in the thunder, spirits in the flame — after...
From Plasma to Game Theory: The Unlikely Journey of an SDE A Self-Referential Equation In 1938, Soviet physicist Anatoly Vlasov was studying plasma — a gas of charged particles so dense that every particle feels an electric field generated by every other. Tracking...
Markov Chains in Supermarket Management The Supermarket as a Stochastic System A supermarket is full of queues, decisions, and uncertainty. Customers arrive unpredictably, spend varying amounts of time at different sections, and leave when they are done. Managing the...
Kelly, the Growth-Optimal Portfolio, and the Benchmark Approach to Option Pricing From Coin Flips to Risk-Neutral Pricing The binomial tree is the cleanest version of the option pricing argument. At each node, the stock moves up by a factor $u$ or down by a factor...
The Bookmaker’s Measure: Kelly, Martingale, and the Price of an Edge The Bettor With a System Every casino and every bookmaker has met this person. They have a system. It is called the Martingale. The logic is airtight: bet one unit, lose, bet two, lose, bet...
The Kelly Criterion — Why the Optimal Strategy Is Never Used A Coin With an Edge Suppose someone offers you a coin. It lands heads 55% of the time. You can bet any fraction of your bankroll on each flip, collect if it lands heads, lose your stake if it lands tails....
Four Historical Decisions an ABM Would Have Changed The Pattern That Keeps Repeating Smoot-Hawley was not a one-off. It was one instance of something older: a decision-maker looking at a complex system through a linear lens, pulling one lever, and watching everything...
The Trade War Nobody Chose: Smoot-Hawley and the Model That Would Have Stopped It A Thousand Economists Walk Into a Room In May 1930, over a thousand American economists signed a petition and sent it to the White House. The letter was unusually direct for academics:...
CAD 530 a Month: What Your Car Insurance Premium Actually Pays For The Bill That Made Me Do Maths A friend of mine recently bought a Ford F-150 in Ontario. Beautiful truck. 75,000 dollars Canadian. He was proud of it for about forty-eight hours, until the insurance...
Langevin Dynamics and Why They Matter in Finance A Particle in a Noisy World In 1908, Paul Langevin wrote down an equation to describe a small particle buffeted by the molecules of the surrounding fluid. Newton’s second law with two extra terms: a friction force...
Two Worlds, One Price: Entropy and the Risk-Neutral Measure Two Probabilities for One Asset In classical physics, probability is a statement about a single world — a particle’s position has one distribution, determined by the Hamiltonian. Finance operates...
Pricing a Gas Liquefaction Plant as a Real Option: An HJB Approach The Limits of Static NPV Infrastructure investment decisions are rarely simple discounted cash flow problems. When you are considering a multi-billion dollar capital commitment whose returns depend on...
Heat Diffusion in an Iron Sphere — A FEniCSx Simulation The Problem An iron sphere with a hot core sitting in air. The core starts at 800°C and the surrounding material begins at ambient temperature. How does heat spread outward? How fast does the core cool? How does...
Solar Magnetohydrodynamics: Solving Magnetic Diffusion in a Sphere The Sun’s Magnetic Field as a PDE Problem The sun is a magnetised plasma — a turbulent, rotating, electrically conducting fluid threaded by magnetic fields of extraordinary complexity. Sunspots,...
Solving the 2D Wave Equation with Spectral Methods The Equation That Models Everything That Vibrates Sound propagating through air, light through space, seismic waves through the Earth’s crust, ripples on a pond — all of these obey variants of the same...
The Square Root That Keeps Rain Positive: CIR Processes in Weather Modelling From Finance to Meteorology The Cox-Ingersoll-Ross process was introduced in 1985 as a model for short-term interest rates. Its defining feature — that it never becomes negative — was useful...
When Neurons Forget: The Ornstein-Uhlenbeck Process in Neuroscience The brain is noisy. Individual neurons fire irregularly, synaptic transmission is unreliable, and the electrical potential across a cell membrane fluctuates continuously even in the absence of any...
What Agent-Based Models See That Regressions Miss The Trap of Spurious Correlation In 1974, Granger and Newbold demonstrated that two independent random walks, when regressed against each other, produce high $R^2$ values and significant $t$-statistics with probability...
The Mean Field Game of Mediocrity Why Capable People Underperform Together In any large institution — a corporation, a civil service, an academic department — a puzzling equilibrium sometimes takes hold. Individually capable people converge on unremarkable output. No...
Agent-Based Models in the Peanut Butter Industry Why Peanut Butter? Peanut butter is a commodity market with brand differentiation, price-sensitive consumers, volatile input costs, and a supply chain that spans farmers, crushers, manufacturers, and retailers. It is...
ABM or MFG? The Cases Where the Answer Is Not Clear Two Frameworks, One Question When you face a system of interacting agents — traders, firms, particles, cells — you eventually have to choose a modeling language. Agent-based models (ABM) simulate each individual...
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.
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.
How Markov chains model checkout queues and customer flow — and what the stationary distribution tells a supermarket manager about staffing and store layout.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
The leaky integrate-and-fire neuron — the workhorse model of computational neuroscience — is an Ornstein-Uhlenbeck process in disguise. Here is why that matters.
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.
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.
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.
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.