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.
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.
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.
