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Topic: Brownian Motion
Replies: 0

 Maury Barbato Posts: 792 From: University Federico II of Naples Registered: 3/15/05
Brownian Motion
Posted: Oct 14, 2012 9:17 AM

Hello,
I was thinking about the definition of Brownian motion,
which is the following.

Def. A standard one-dimensional Brownian motion is a continuous adapted process such that
1) B_0 = 0 a.s.
2) for every positive integer n, and every
0 <= t_1 <= ... <= t_n, the increments
B_{t_n} - B_{t_{n-1}}, ..., B_{t_2} - B_{t_1}
are independent and normally distributed with mean 0
and variances respectively t_{n} - t_{n-1}, ...,
t_2 - t_1.

I think that the requirement in 2) that we can choose
every positive integer n is essential. For example,
I think it is not enough to require that 2) holds
only for n=2, but I could not give a counterexample.