Stochastic differential equation
Quantitative Finance glossary · Stochastic Calculus & Continuous Time Finance
Definition
Equation of form dX_t = μ(t, X_t),dt + σ(t, X_t),dW_t describing the infinitesimal evolution of X_t as a deterministic drift plus a Brownian diffusion. Lipschitz/linear-growth conditions on μ, σ guarantee strong existence and uniqueness of solutions (Itô 1951). Numerically integrated by Euler-Maruyama (O(√(Δ t)) strong convergence) or Milstein scheme (O(Δ t), requires partial_x σ). Forms with dN_t jump terms or dB_t^H (fractional Brownian) extend the framework to jump-diffusion and rough-vol models.
Stochastic differential equation in plain English
This is the math for describing something that drifts along a trend while also getting jostled by constant random noise — like a stock price that generally rises but wiggles unpredictably every second. The equation splits change into two parts: a predictable "drift" (μ) pushing it in a direction over time, and a random "diffusion" (σ) scaling the size of the jitter, where the jitter comes from Brownian motion (the same aimless staggering you'd see in a speck of dust bouncing off air molecules). You usually can't solve it with a neat formula, so computers step through tiny time slices adding a bit of drift plus a bit of random shock — the simple Euler-Maruyama recipe, or the more accurate Milstein one. It's the workhorse behind nearly every model of how prices, interest rates, and volatility evolve in continuous time.
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