Research desk
Optimal Control & Decision
What to do under uncertainty: optimal control and HJB, mean-field games, agent-based models, irreversible decisions, betting, lotteries and insurance.
The desk works on what to do under uncertainty: Hamilton–Jacobi–Bellman equations, dynamic programming, optimal stopping and singular control, and the games that arise when several players optimise at once — Dynkin games, wars of attrition, and mean field games with their coupled Hamilton–Jacobi–Bellman and Fokker–Planck systems. Applications run from real options and irreversible investment to dividend policy, leverage caps, battery and turbine control, and monetary policy; agent-based models cover the cases where equations give out. Betting, lotteries, insurance and the Kelly criterion belong here too.
Does a Nobel Formula Beat Dividing by N?
A runnable notebook. The Talmud says to divide your money in three; Markowitz's Nobel formula weighs means and covariances. On 1,000 random portfolios of 25 S&P 500 stocks the textbook formula was wiped out in 862 and its long-only form beat 1/N in 149. Fed…
What Does a Kaggle Prize Pay?
A runnable notebook on Kaggle's own record, Meta Kaggle. An entry in a prize competition is a lottery ticket paid for in hours. Most Kaggle competitions pay no cash; across the 649 that did, the pot came to 48.62 dollars an entry and a team's…
The Bet Samuelson’s Colleague Refused
A runnable notebook. In 1963 Paul Samuelson told of a colleague who refused one coin toss paying 200 dollars against 100 but would take a hundred. Over 98 years of the S&P 500 with dividends, a daily look shows a loss 46% of the time,…
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…
What Does a Nonlinear Fokker–Planck Equation Buy a Mean Field Game?
Mean field games almost always pair the agents' Hamilton–Jacobi–Bellman equation with a Fokker–Planck equation linear in the density: the agents' noises are independent, every interaction sits in the cost, and the agents never touch. This discussion paper asks what each kind of nonlinearity in the…
Both Sides of the Policy Get Richer
A runnable notebook. In 1738 Daniel Bernoulli's merchant Caius faced an 800-ruble premium on a cargo whose expected loss was 500, and Bernoulli found that a merchant worth less than 5,043 rubles should buy it and an insurer worth more than 14,243 should sell it.…
Is a Lottery Ticket Ever Worth Buying?
A runnable R notebook. From 2004 to 2012 the Massachusetts lottery game Cash WinFall had a roll-down rule that made a two-dollar ticket worth more than two dollars on 44 draws. The value of a ticket on those nights is one line, the fixed prizes…
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…
Littlewood’s Rule for a Home Battery
Ontario's Ultra-Low Overnight electricity plan charges 3.9 cents a kilowatt-hour overnight and 39.1 cents between 4 and 9 PM on weekdays, and home batteries are sold on that tenfold spread. This paper asks how such a battery should be run when the price is set…
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…
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…
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…
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…
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…
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…
A Leverage Cap on an Inconsistent Investor
The mean–variance criterion does not satisfy Bellman's principle, so "optimal" splits into three incompatible strategies: a pre-commitment plan that requires a commitment device, a naive agent who re-solves continually and follows none of his plans, and an equilibrium from which no future self wishes to…
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…
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…
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.
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.…
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…
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.
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.
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.
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.
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.
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.
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…
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.
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.
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.
Last Passage Times, the Azéma–Yor Martingale, and Optimal Prediction of the Maximum
We study last passage times of standard Brownian motion and their role in the optimal prediction of the running maximum. Using explicit distributional formulas from Borodin and Salminen's Handbook of Brownian Motion, we characterise the law of the last passage time ga = sup{t ≤ 1: Bt = a} and connect it…
A Huggett-Moll Mean Field Game with Mean-Reverting Kou Jump-Diffusion Productivity
We formulate a Huggett-Moll mean field game in which agents' labor productivity follows a Kou double-exponential jump-diffusion, capturing asymmetric income shocks: sudden collapses and rare windfalls superimposed on geometric Brownian motion. Working in log-productivity space x = log z, the multiplicative jump structure becomes additive, and both…
Wealth Stratification, Norm Contagion, and Institutional Capture: An Agent-Based Model of Endogenous Accountability Collapse
We introduce an agent-based model of N interacting agents, each carrying wealth Ri(t) evolving under Kou pure-jump dynamics and an accountability level Ωi(t) ∈ [0,1]. Three interlocking feedback loops drive the system: Loop 1 links wealth rank ri(t) to an erosion rate α(ri) = α0 + α1 riγ, accelerating accountability…
Accountability Collapse under Wealth Growth: A Stochastic Control Problem with Integral Equations
We formulate the erosion of social accountability Ω(t) under growing private wealth R(t) as a stochastic optimal control problem on ℝ+. The wealth process follows a pure Kou double-exponential jump model with no diffusion and no drift: two independent compound Poisson processes with rates…
The Controlled Symbol: Pseudo-Differential Operators, HJB Duality, and Spatially Varying Regularity
We develop a rigorous treatment of Feller processes as pseudo-differential operators and apply the resulting symbol calculus to stochastic optimal control. Starting from the Courège–Lévy–Khintchine representation theorem, we define the symbol q(x,ξ) of a Feller generator as the position-dependent analogue of the Lévy exponent, and…
Optimal Stopping and the Snell Envelope
We study the optimal stopping problem in continuous time, where an agent chooses a stopping time to maximise the expected value of a payoff process, following the classical framework of Snell (1952) and its continuous-time extension via the Doob–Meyer decomposition. The value function is characterised…
The Two-Player War of Attrition on a Bivariate Diffusion: A Free Boundary PDE Approach
We study a two-player war of attrition in which each firm's profitability evolves as an independent geometric Brownian motion, so the state space is two-dimensional. The game is formulated as a Dynkin stopping game whose equilibrium characterises a pair of free boundaries Γ1 and Γ2 …
The Three-Player War of Attrition as a Dynkin Game
We study the three-player war of attrition as a Dynkin game in continuous time, driven by a geometric Brownian motion representing market demand. Each of three symmetric firms controls a stopping time; the last firm to exit captures the entire market. Because the game is…
Optimal Dividends with a Resurrection Option in the Cramér–Lundberg Model
We study the optimal dividend barrier problem for the Cramér–Lundberg surplus model when the firm's owner holds a one-shot resurrection option: upon ruin, the owner may pay a fixed cost R to restart operations at a prescribed level x0. The W(q) scale function, characterised…
A Stationary Mean Field Congestion Game on a Torus
We study a stationary discounted mean field game on the one-dimensional torus T=[0,1], in which a continuum of identical agents controls a Brownian drift subject to a local congestion cost and a sinusoidal external potential f(x)=Asin(2π x). The equilibrium pair (V*,m*) satisfies a coupled…
Dividends When Capital Is Slow: de Finetti’s Problem with a Sticky Boundary
Dividend models give a firm at zero surplus two fates: it is ruined, or capital arrives instantly and reflects it back into business. Neither is what happens when capital must be raised, which takes time, and while it is being raised the firm can neither…
The Price of a Yaw: Optimal Deadbands and Filters for Wind-Turbine Yaw Control, Calibrated on SCADA Data
A wind turbine must keep its rotor facing a wind whose direction never stops moving, and every yaw that realigns it wears the yaw system. Using a year of 10-minute records from the six turbines of the Kelmarsh wind farm, this paper measures what sets…
Projected SOR, Penalty and Policy Iteration for Real-Options Variational Inequalities
An option to invest whose payoff depends on two random factors is valued by a variational inequality; after a monotone discretisation it is a linear complementarity problem with an M-matrix, which has one solution. This report compares projected SOR, the penalty method, policy iteration and,…