← Stochastic Differential Equations

Brownian increments

Over a time interval of length \(\Delta t\), the Brownian increment is

\[\Delta W=W(t+\Delta t)-W(t).\]

Its distribution is

\[\Delta W\sim N(0,\Delta t).\]

Equivalently,

\[\Delta W=\sqrt{\Delta t}\,Z,\qquad Z\sim N(0,1).\]

The factor \(\sqrt{\Delta t}\) appears because multiplying a standard normal variable by it gives variance \(\Delta t\).

Key idea. Brownian increments have standard deviation \(\sqrt{\Delta t}\), not \(\Delta t\).