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Maury Barbato
Posts:
789
From:
University Federico II of Naples
Registered:
3/15/05
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Brownian Motion
Posted:
Oct 14, 2012 9:17 AM
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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.
What do you think about?
Thank you very much.
My Best Regards,
Maury Barbato
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