Question

How to prove a Brownian Motion Process {X(t),t>=o} that its X(t) is normally distributed with mean 0 and variance σ²t using Central Limit Theorem?

Answer 1

motion an impostant continuous Boo Dian time stochastic psocess that Sesves as tontinous time onalog to simple sand om walk on the one hand and shaže fundamental psocessonthe othes hand summet γι C opesties with he Pois8 on counting Thsoughout, we use the followin g notation fos the seal numbes s the pon negative seal numbess the intešzess and th e mon- negati ve inteqgess spectively def def 2 01, 2

ue obsesye that the vectos( BB4n) has a multiva81ate no mai distsi bution because the event can be ve- usitten in tesms of indepen dent in csement events Bltn)-un-) yiclding the joint density oP (et)B (th) as uhese is the densithe Ntot) dlistsibution IS The finite dimensioncal dists ibutio ns of BM ase -thus multivasiate nosmal C Co) naussian and an example of , a Gaussian pocess, a pboics Continuous sample hat Po th i cwhich the Finite dimensionat distsibutions ae multivasiate nosmal that is fos any fixed choice oe n time points nosmal

sinte a multivasiate nosmal distši bution completely detesmined by its mean and Covasian,e posametess, we conclude that a Graussian Psoces s is completely detesmi ned by its mean an d def var (Bis)) to Thus standasd BM is the unique 6aussian pšocess wi Similalg BM uwith vasiance and drift u,x (4)