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. Missing the 10 Best Days
Fund companies print it every year: miss the 10 best days and your money is cut in half. On the S&P 500 with dividends since 1990, a dollar became 44.6, or 19.8 without its 10 best days. Miss the 10 worst too and it became 47.6, more than staying in. And 7 of the 10 best days came within a month of one of the worst.
Read more →2. Right 50.75% of the Time
Medallion, the fund that made 66% a year, was reportedly right on just 50.75% of its trades, and it placed 150,000 to 300,000 of them a day. The arithmetic shows how that tiny edge becomes a fortune: at least 4,505 independent bets a year, a correlation between them below 1 in 4,504, and a stake that sets the chance of ever losing half the money whatever the edge: 50% at full Kelly, 12.5% at half.
Read more →3. Should You Wait for a Dip?
Waiting for a 10% dip before buying the S&P 500 left the waiter behind in 72% of the months since 1990, and a short theorem says why: when the market beats cash, every waiting rule loses on average. For a price that comes back, waiting is right. The textbook answer for one trade says wait for a fall of more than two typical swings; a trader who keeps trading should buy at less than one.
Read more →4. The 1966 Retiree
A retiree of 1966 who drew 4% a year from the stock market ran out of money in 1992, though the market earned 4.87% a year after inflation; the same thirty years in reverse order would have left 2.34 times the starting sum. Across 126 retirements since 1871, 4% failed three times and 5% failed 24 times, and Dufresne's identity gives the chance that a withdrawal ever runs out in closed form.
Read more →5. The Cell That Solves Bellman’s Equation
A bacterium shifted to better food must decide, protein by protein, whether to build ribosomes or the enzymes that feed them. Posed as optimal control, Bellman's equation says the answer is all or nothing, switched along a single curve. E. coli has no brain, but its alarm molecule ppGpp switches the same way, and gets 98% of what is possible.
Read more →6. The Catch That Emptied the Grand Banks
In 1992 Canada closed the northern cod fishery, its stock at about 1% of what it had been. A fixed quota at the maximum sustainable yield sounds prudent, but its balance point is a cliff edge: with good and bad years, 93% of model stocks fished that way collapse within 50 years. The optimal rule, solved exactly, takes more in good years, nothing in bad ones, and never collapses from fishing.
Read more →7. When to Shoot
Two duellists walk towards each other, one bullet each. Fire too soon and you will probably miss, and a miss is fatal. In 1949 David Blackwell worked out the answer in a day: fire when your two chances add up to one, the good shot and the bad shot at the same instant. Then he made the pistols silent, and the only safe strategy became a roll of the dice.
Read more →8. The Game That Remembered
Roulette cannot be beaten because the wheel forgets every spin. A deck of cards does not forget. In 1961 a young mathematician proved it, took the proof to Reno and won $11,000 in a weekend. Then he found the harder truth: with a real edge you can still go broke, and betting twice the right amount earns you exactly nothing.
Read more →9. The System That Could Not Lose
Dostoevsky wrote home from Wiesbaden that he had found the secret of roulette: you just can't lose. Then he signed away his future, gambled the advance, and dictated a novel in 26 days to save himself. The doubling system every gambler knew wins 39 nights out of 40 — and loses exactly the house edge anyway.
Read more →10. The Road That Made Everyone Late
Close a busy road and the traffic gets faster. Cities have done it and been surprised. The reason is arithmetic, it needs every driver to be perfectly selfish and perfectly informed, and in the smallest version a free new road costs everybody fifteen minutes. Then it turns out the most destructive road of all is not the free one.
Read more →11. Everybody Wants the Cheap Plug
Ten thousand vehicles, two cheap places to charge and one hour. A mean field game — Hamilton–Jacobi–Bellman backwards, Fokker–Planck forwards — solved on a torus and animated, and the 1.60% that nobody's mistake accounts for.
Read more →12. The Cube Root That Runs the Treasury
How much cash should a company keep? If its daily cash flow is a random walk and every transfer costs a fee, the best rule is a band: do nothing inside it, and jump back to a return point a third of the way up. The return point is a cube root, which makes the rule remarkably forgiving.
Read more →13. De Rerum Electione: The Choice That Is Not a Swerve
A sequel in verse to De Rerum Stochástica. Lucretius put free will in the random swerve; Cicero called that childish, Laplace took the swerve away, and Born gave chance back to physics. The mathematics of control puts freedom somewhere else: in the steering, not the noise.
Read more →14. 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 →15. 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 →16. 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 →17. 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 →18. 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 →19. 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 →20. 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 →21. 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 →22. 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 →23. 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.
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