Essays
These are our essays. They showcase applied mathematics across a wide span of subjects, from physics and engineering to finance, history and medicine: the models, the methods that solve them, the code that runs them and the algorithms behind them, often with a live model you can try for yourself. Each essay starts from a question and works it through to a number. Nothing here is a product or advice. The code we publish as Jupyter notebooks is under Notebooks, the live boards that each answer one practical question are under Tools, and the research papers are under Research Papers. Every essay belongs to one of our five research desks; the buttons below show one desk's essays at a time.
1. Reject Three Out of Four
The random walk that estimates central-bank models, dated the start of the HIV pandemic and measured the age of the universe works best when it rejects about three proposals in four. In many dimensions the walk becomes a Langevin diffusion, and 23.4% is where that diffusion runs fastest; a walk that accepts 90% needs 23 times the steps.
Read more →2. The Barrier That Comes Early
A price starts at $100 and the average wait until it first reaches $120 is nine years. Yet there is a 40% chance it gets there within a year, and the most likely moment is three months away. Under geometric Brownian motion the whole distribution of that wait is one closed formula, which prices an up-and-out call and a clause in a project contract alike.
Read more →3. The Urn That Decides Once
An urn with one red ball and one black. Draw one, put it back with a friend of the same colour, forever. The share of red settles — and it settles on a number drawn uniformly from nought to one, so there is no typical outcome at all. Then de Finetti removes the mechanism entirely, and one exponent decides whether the rich getting richer ends in a monopoly or in a permanent accident.
Read more →4. The Bird Does Not Come Home
A drunk man and a drunk pigeon leave the same pub with all the time in the world. The man is certain to find his way back. The pigeon manages it 34% of the time and otherwise never. One extra dimension turns a certainty into a bet you mostly lose — and then the sky turns out to have a ceiling.
Read more →5. 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 →6. 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 →7. 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 →8. 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 →9. 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 →10. The Lead That Never Changes Hands
A fund manager with no skill spends at least 80% of twenty years ahead of the index with probability 29.5%, and a record that looks fair, ahead 40 to 60% of the time, turns up only 12.8% of the time. Lévy's arcsine law explains why, and why the same curve decides when the lead last changed hands.
Read more →11. The Return That Takes Four Centuries
A share with 20% volatility and a 5% expected return gives away its volatility in a single day of one-minute prices, to within 0.72 of a point. Its expected return needs 400 years of data to be known to within one point. Quadratic variation and Girsanov's theorem explain why, and why sampling more often cannot help.
Read more →12. De Rerum Stochástica: Lucretius’s Random Swerve and the Birth of SDEs
Epicurus added a random, uncaused swerve (Lucretius would call it the clinamen) to save atomism from determinism. Two thousand years later, it became the diffusion term in every SDE ever written.
Read more →13. 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 →14. 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 →15. 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.
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