
Brownian motion
Let $\displaystyle {B}(t)$ be a Brownian motion. Show that the following processes are Brownian motions on [0,T].
1. $\displaystyle {X}(t)={B}(t)$.
2. $\displaystyle {X}(t)={B}(Tt)  {B}(T),$ where $\displaystyle {T}<+\infty$.
3. $\displaystyle {X}(t)=c{B}(t/{c}^2),$ where $\displaystyle {T}\leq+\infty$.
4. $\displaystyle {X}(t)=t{B}(1/t),t>0, $ and $\displaystyle {X}(0)=0$.
Who ever can answer all of these is legendary. Any help will be greatly appreciated. Thanks.

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