Brownian motion
Quantitative Finance glossary · Stochastic Calculus & Continuous Time Finance
Definition
Continuous-time stochastic process W_t with (i) W_0 = 0, (ii) independent increments, (iii) W_t - W_s sim mathcalN(0, t-s), and (iv) continuous but almost-surely nowhere-differentiable paths. Quadratic variation langle W rangle_t = t — the property that makes Itô calculus distinct from ordinary calculus, since (dW_t)^2 = dt does not vanish. The basic noise driver of every continuous-time finance model (GBM, Vasicek, Heston, HJM).
Brownian motion in plain English
Imagine a speck of dust floating on water, getting knocked in random directions thousands of times a second so its path is an endless jittery zigzag. Brownian motion is the clean mathematical version of exactly that: a path that starts at zero, wanders with no memory of where it's been, and is so infinitely jagged that you can never say which way it's 'currently heading.' It is the fundamental source of randomness that nearly every model of moving prices and interest rates is built on top of. Its key quirk — that the wiggles are so violent the squared movement over a tiny time step dt doesn't shrink away to nothing — is precisely why finance needs its own special calculus (Itô calculus) instead of the ordinary kind.
Learn these first
Related Quantitative Finance terms
Quantitative Finance practice questions · Quantitative Finance flashcards · Quantitative Finance study guide