# Math Help - Brownian motion

1. ## Brownian motion

Let ${B}(t)$ be a Brownian motion. Show that the following processes are Brownian motions on [0,T].

1. ${X}(t)=-{B}(t)$.

2. ${X}(t)={B}(T-t) - {B}(T),$ where ${T}<+\infty$.

3. ${X}(t)=c{B}(t/{c}^2),$ where ${T}\leq+\infty$.

4. ${X}(t)=t{B}(1/t),t>0,$ and ${X}(0)=0$.

Who ever can answer all of these is legendary. Any help will be greatly appreciated. Thanks.

2. All your questions regarding Brownian motions is simply a matter of applying and checking the definition.

Definition of Brownian motion:
A continuous random process X(t) is a (standard) Brownian motion if it satisfies

1. X(0) = 0
2. Independent increments
3. Increments are normally distributed with mean zero and variance equal to the time increment.