a stochastic differential equation
An ordinary differential equation describes how something evolves smoothly: it gives the rate of change at each instant, like 'population grows at 3 percent per year', and you integrate to get the trajectory. A stochastic differential equation (SDE) describes evolution that is partly steered and partly buffeted by random noise. It is the natural model for anything that has a tendency PLUS jitter: a stock price drifting up while shaking, a particle pulled toward a centre while kicked around, a population trending while randomly disturbed. An SDE is the equation of motion for a noisy world.
The standard form has two pieces: dX = a(X, t) dt + b(X, t) dB. The first term, a(X, t) dt, is the drift — the smooth, predictable tendency, the average direction of motion. The second term, b(X, t) dB, is the diffusion — the random kick, a Brownian increment scaled by a volatility b(X, t) that can itself depend on where you are and when. Read the whole thing as: 'in the next instant, X moves by an average amount a dt PLUS a random shake of typical size b times sqrt(dt)'. Crucially, this is shorthand. Because dB is not a real differential, the only rigorous meaning of an SDE is its integrated form X(t) = X(0) + 'integral of a ds' + 'integral of b dB', where the last term is an Ito integral. You solve SDEs not by ordinary integration but with Ito's lemma — guess a function of B and use Ito's rule to check it produces the right drift and diffusion.
SDEs are the working language of quantitative finance, physics, biology, and engineering. Geometric Brownian motion (the Black-Scholes stock model), the Ornstein-Uhlenbeck process (mean reversion), and population and epidemic models with noise are all SDEs. Each links, via the Feynman-Kac formula, to a partial differential equation for its probability density. Two honest caveats: first, an SDE only makes sense paired with an interpretation of its stochastic integral — the Ito reading (default in finance) and the Stratonovich reading (common in physics) give genuinely different solutions, so you must say which you mean. Second, this is the deep end of probability; existence and uniqueness of solutions need conditions (typically Lipschitz coefficients), and most SDEs have no closed-form solution and must be simulated.
A noisy savings account: dX = r X dt + sigma X dB. The drift r X dt grows the balance at rate r (compound interest), while sigma X dB adds market jitter proportional to the balance. This particular SDE has a closed-form solution — geometric Brownian motion — found by applying Ito's lemma to log(X).
An SDE = predictable drift a dt plus random diffusion b dB; its only rigorous meaning is the integrated (Ito) form.
An SDE is meaningless until you fix the integral convention: the Ito and Stratonovich readings give different solutions. 'dX = a dt + b dB' is shorthand for an integral equation, not a literal derivative.