Posers and Puzzles
30 Sep 04
01 Oct 04
Originally posted by tejoIs time discrete or continuous? Could you define Brownian motion mathematically?
I could use some help with this exercise I just got.
Prove that with chance zero that limit for t->infinite (Brownian motion at time t)/(sqrt (t))=0.
I hope you understand what I mean and that you can help me.
Thanks,
Sander
01 Oct 04
Originally posted by AcolyteI hope I will explain this well.
Is time discrete or continuous? Could you define Brownian motion mathematically?
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.
01 Oct 04
1 edit
Originally posted by tejoI understand the question now. I think the following will work: Bt (for t an integer) can be turned into a discrete summation; conditions 1. and 4. imply that the summands are iid with variance 1, and then a Central Limit Theorem does the rest.
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 indep ...[text shortened]... 1.
The Brownian Motion is also called Wiener process I believe.
04 Oct 04
Originally posted by AcolyteThank you very much Acolyte. It worked really well.
I understand the question now. I think the following will work: Bt (for t an integer) can be turned into a discrete summation; conditions 1. and 4. imply that the summands are iid with variance 1, and then a Central Limit Theorem does the rest.