STOCHASTIC PROCESSES 15
Brownian Motion
Brownian motion is a continuous stochastic process whose increments are independent, normally distributed and scaled by elapsed time.
Definition
A standard Brownian motion \(\{W_t:t\ge0\}\) satisfies \(W_0=0\). For \(0\le s\lt t\),
\[\boxed{W_t-W_s\sim N(0,t-s)}\]
1. Independent increments
Brownian increments over non-overlapping intervals are independent.
2. Stationary increments
\[W_{t+h}-W_t\sim N(0,h).\]3. Brownian increment
\[\Delta W=W(t+\Delta t)-W(t)\sim N(0,\Delta t).\]\[\boxed{\Delta W=\sqrt{\Delta t}\,Z,\qquad Z\sim N(0,1)}\]
4. Why the square root appears
\[\operatorname{Var}(\sqrt{\Delta t}Z)=\Delta t.\]5. Continuous does not mean smooth
Brownian paths are continuous but almost surely nowhere differentiable.