# Brown motion process

1. How many types of Brownian motions do we have?

2. And When somebody says "Brownian motion generated by a random walk", which one of those types of Brownian motion it is?

3. How do we judge if a stochastic process is a Brownian motion? IS it that any process which has the characteristics mentioned below is a BM process?

My current knowledge is this: If $X_t$ be a BM process then

1. $X_0=0$

2. and $X_t$ has homogeneous stable independent increment.

3. $X_t$ is normally distributed $N(0,c^2t)$

If $c=1$ then we have a Standard BM and we show it by $B_t$.

BM is not a stationary process, but its increment is stationary.

• What counts as "a type" of Brownian motion for you? – Henning Makholm Dec 25 '16 at 9:53
• Only if in knew ... We have this question in the book that is asking us to compare two Brownian motion. I think (not sure) maybe it means like: Standard BM, BM with drift, Geometrical BM, Brownian bridge, 2-D BM? maybe? I have no idea...just guessing – Tayebe Dec 25 '16 at 10:18