Originally posted by Acolyte
Is time discrete or continuous? Could you define Brownian motion mathematically?
I hope I will explain this well.
Bt:=integral from 0 to t Ns ds.
Ns is a completely random function of t. In other words, the continuous-time analogue of a sequence of independent identically distributed random variables.
1.Ns is independent of Nt for t != (not equal) s
2.The random variables Ns (s>=0) all have the same probability distribution u
3.Expectancy of Ns = 0 (E(Nt)=0)
So Bt has the following requirements:
1.For any 0=t0<=t1<=...<=tn the random vaiables Bt(j+1) - Bt(j) are independent (j=0,...,n-1)
2.Bt has stationary increments.
3. E(Bt)=0 for all t>=0
4. E((B1)^2)=1 also true E((Bt)^2)=t
5. t->Bt is continuous a.s. with probability 1.
The Brownian Motion is also called Wiener process I believe.