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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 onedimensional 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_{n1}}, ..., B_{t_2}  B_{t_1} are independent and normally distributed with mean 0 and variances respectively t_{n}  t_{n1}, ..., 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



