- 01 Oct '04 16:36

Is time discrete or continuous? Could you define Brownian motion mathematically?*Originally posted by tejo***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 17:02

I hope I will explain this well.*Originally posted by Acolyte***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 23:06 / 1 edit

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.*Originally posted by tejo***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 11:51

Thank you very much Acolyte. It worked really well.*Originally posted by Acolyte***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.**