How to check if a process is brownian motion or a martingale?

[CorkActuary]

A little stuck on this topic. The lectures I have are not very clear and I don't have a copy of the CT8 Acted Notes

For example:
1) Is Bt^2 - t a martingale?
2) Is (1/c)Bct a Brownian Motion?

 

[Oxymoron]

A little stuck on this topic. The lectures I have are not very clear and I don't have a copy of the CT8 Acted Notes

For example:
1) Is Bt^2 - t a martingale?
2) Is (1/c)Bct a Brownian Motion?

Start off by expanding any Brownian motion term as increments :-
Bt^2
= (Bt - Bs + Bs)^2
= (Bt-Bs)^2 + Bs^2 + 2*(Bt-Bs)*Bs

Now take expectation :-
E(Bt^2)
= E[(Bt-Bs)^2 + Bs^2 + 2*(Bt-Bs)*Bs]
=E[(Bt-Bs)^2] + E[Bs^2] + 2*E[(Bt-Bs)]*E[Bs] :- independence property of Brownian motion increments
= (t-s) + s + 0
= t

E(Bt^2) = t
=> E(Bt^2 - t) = 0

Hence it's a martingale.

For part II, apply mean, variance and covariance properties and check the results. Properties of Bt should be the same as (1/c)Bct