Research desk
Stochastic Analysis
How random processes behave — Itô calculus, diffusions, local time, Fokker–Planck — in theory and in application.
The desk studies how random processes behave, in theory and in application: Itô calculus, diffusions and their boundaries, local time and Tanaka's formula, skew Brownian motion, Ray–Knight laws, diffusions with resetting, and the Fokker–Planck equation. Applied, the same tools explain volatility drag in stock returns, false correlation in random matrices, Ornstein–Uhlenbeck and CIR models in neuroscience and weather, and the numerics that get them right: Euler–Maruyama against Milstein, Brownian-bridge corrections for barriers, and the diffusion limit behind Metropolis tuning.
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…
63% of Stocks Lost to the Average Stock
A runnable notebook. A dollar in each of 459 S&P 500 stocks for five years: the average stock turned $1 into $2.00, the median stock into $1.73, and 288 of the 459, 62.7%, finished below the average. Compounding alone does it, by an amount the…
The Noise That Looks Like Correlation
A runnable notebook. Two hundred independent assets still show correlations; the Marchenko–Pastur law says how wide the band of noise eigenvalues is, a planted market shows which eigenvalues rise above it, and a minimum-variance portfolio built on the raw matrix promises less risk than the…
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…
The Contact Field: Ray-Knight Laws and Reaction in a Distributed Reactive Medium
A particle diffusing through a reactive medium reacts not on first contact but when the contact it has accumulated crosses a threshold, so the quantity that decides its fate is a random field: the contact accumulated at every level it can react at. For one-dimensional…
Skew Brownian Motion: One Biased Site and the Asymmetry It Buys
Skew Brownian motion is ordinary Brownian motion everywhere except one point: at the origin each excursion chooses the positive side with probability α and the negative side with probability 1-α. Nothing else is altered — no drift, no varying diffusivity, no boundary. This note…
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…
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…
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.
Diffusions with Stochastic Resetting
We study diffusions subject to Poissonian resetting: at rate r the process is returned instantaneously to a fixed point x0. Two consequences separate this from the diffusions of classical stochastic analysis. First, the reset generator Lr f = Lf + r[f(x0) - f(x)] is non-local and, even when L is self-adjoint…
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.
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.
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.
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.
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,…
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…
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.
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.
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.
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.
The Square-Root Wall
Stop a Brownian path the first time it reaches c standard deviations of its own age, |Bt| = c√1+t. In the clock s = log(1+t) the wall becomes the fixed interval (-c, c) for an Ornstein–Uhlenbeck process, and every moment of the wait follows: 𝔼[(1+τc)p] = 1/M(-p, tfrac12, c2/2), with M Kummer's…
Brownian Local Time, Tanaka’s Formula, and the Quantum Delta Potential
We construct Brownian local time Ltx as the density of the occupation measure of standard Brownian motion and establish three foundational results: the occupation time formula, Tanaka's formula extending Itô's lemma to |Bt - a|, and Lévy's representation theorem identifying Lt0 in distribution with |Bt|.…
The Distribution of Growing Perpetuities and the Gordon Growth Formula
A cash flow growing at expected rate g with volatility σ and discounted at r > g for ever is worth S0/(r-g) by the Gordon growth formula. That number is the mean of the present value, not its typical size. The present value is an exponential functional…