Stochastic Differential Equations

Many systems remain uncertain even after their initial state and parameters are fixed because unresolved effects continue to perturb their evolution. A stochastic process is a collection of random variables indexed by time. A stochastic differential equation (SDE) specifies how such a process evolves by combining a drift, its systematic local trend, with a diffusion term, which controls its local variability. Brownian motion, a continuous stochastic process with independent Gaussian increments, supplies the random fluctuations. Brownian paths are continuous but have no ordinary time derivative, so the random term requires a new interpretation of integration.

The treatment assumes ordinary differential equations and numerical time integration; differential and integral calculus, including partial derivatives, gradients, and Hessians; linear algebra; elementary probability, including random variables, independence, conditional Gaussian distributions, expectation, variance, and covariance; and likelihood-based parameter estimation and numerical optimization.

We first define Brownian motion. A sigma-algebra represents the events that can be resolved from available information, while a filtration records how that information grows with time and enforces causality. We use these concepts at this intuitive level; the measure theory needed to prove that the constructions are well defined is outside our scope. The Itô integral then gives meaning to integration against Brownian motion, Itô’s formula supplies the corresponding chain rule, and Euler–Maruyama provides a numerical transition model. The examples construct Brownian paths and fit stochastic exponential growth with multiplicative noise; a companion notebook illustrates mean reversion with the Ornstein–Uhlenbeck process. Combined with an initial-state distribution and a measurement model, the numerical transition defines a state-space model for hidden-state and parameter inference.