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
