Brownian Motion - Independent Increments

Brownian motion Y(t) has independent increments: the differences Y(t4) - Y(t3) and Y(t2) - Y(t1) are independent of one another, so long as t1 < t2 < t3 < t4.

To illustrate this visually, we sample a Brownian motion simulation and compute increments

incr1 = Y(t2) - Y(t1), incr2 = Y(t4) - Y(t3), ..., incr1000 = Y(t2000) - Y(t1999).

We must take the ti so ti+1 > ti; if the sampling increments overlap, we should not expect independence.

Then we plot the points

(incr2, incr1), ..., (incr1000, incr999).

If the increments are independent of one another, the points should lie in an approximately circular cloud, denser near the center. Here is an example plot.

Return to Mathematical Properties of Brownian Motion.