Brownian Motion
Brownian motion (Wiener process) is the continuous-time random walk with independent Gaussian increments — the backbone of Black–Scholes, Ito calculus, and most diffusion models.
Definition
W_0 = 0, independent increments, W_t − W_s ~ N(0, t−s), continuous paths. It is nowhere differentiable. Geometric Brownian motion is the exponential of a drifted W and is the BS stock.
Why it matters
If your model is a diffusion, you have assumed Brownian (or a time-change of it). Jumps are the thing it cannot do without extra process.
Case
Black–Scholes is GBM. Monday gaps and 1987 are why people add jumps or stochastic vol — Brownian paths do not jump.
How to read it
Ask whether the application needs continuous paths. If the risk is a gap, Brownian is the wrong primitive.
Key papers
Landmark references with real publisher or arXiv links — not token-overlap noise.
- The Theory of Brownian MovementEinstein · 1905 · Annalen der PhysikPhysical Brownian motion — the ancestor of the Wiener process in finance.