Prev: Mobius Band as Bundle Over S^1: Trivialization Containing (1,0) inS^1
Next: Proving Two Statements To Be Equal
From: Olumide on 24 Feb 2010 04:15
Would someone kindly explain to me why is white noise considered to be
the the derivative of Brownian motion?

I know what white noise is, and I know what Brownian motion is. I just
don't know how they relate.

Thanks,

- Olumide
From: µ on 24 Feb 2010 05:20
Olumide a �crit :
> Would someone kindly explain to me why is white noise considered to be
> the the derivative of Brownian motion?
>
> I know what white noise is, and I know what Brownian motion is. I just
> don't know how they relate.

The variation of Brownian motion between instants t and t+dt is normal
of variance dt and, moreover, independant of the variation over antoher
time segment [s,s+ds].

--
M�
From: Jon Slaughter on 24 Feb 2010 05:25
Olumide wrote:
> Would someone kindly explain to me why is white noise considered to be
> the the derivative of Brownian motion?
>
> I know what white noise is, and I know what Brownian motion is. I just
> don't know how they relate.
>

http://en.wikipedia.org/wiki/Brownian_noise

"A Brownian signal is expressed mathematically as the integral of a white
noise signal"

compare the spectrums and the random process properties and you'll see that
they are easily related. You can relate just about any two functions through
their spectrums so it's not that big a deal. It just so happens that for
brownian motion it is a simple expression. Well, it's not arbitrary as the
processes are related in a somewhat logical way.


e.g., the spectrum of white noise is a constant and the spectrum of brown
noise is a constant/w. By the property of the fourier transform we can
easily see how the spectrums are related since the 1/w comes from
integrating. If it were a w it would be from differentiating, w^2 would be
from differentiating twice, and some rational function would result in some
int and derivative relations.


From: brieucs on 24 Feb 2010 05:42
hi,

> Would someone kindly explain to me why is white noise considered to be
> the the derivative of Brownian motion?
>
> I know what white noise is, and I know what Brownian motion is. I just
> don't know how they relate.

a trajectory of a brownian motion is
an almost surely continuous function,
but a.s. not differentiable as a function
of time;

the derivative of a trajectory of a B.M.
(function of time), can be interpreted as
a distribution (Schwartz); this distribution
is the *white-noise* associated to the
trajectory;

this leads to a clever theory, involving
Gelfand-Triples, and "huge" spaces;

in the same time, physicists can present
some simple notions based on discretisations
of time, or audacious but not already
rigorous considerations (Feynmann);

I guess you refer to discrete White Noise;
"delta B / delta t" makes sense in the
discrete way, during (t, t + delta t);
it gives an idea of the white noise;

cf: Stochastic Analysis; Ito, Hida,
Malliavin, Hui Hsiung Kuo (25 years
of white noise), Oksendal, Albeverio,
....


 | 
Pages: 1
Prev: Mobius Band as Bundle Over S^1: Trivialization Containing (1,0) inS^1
Next: Proving Two Statements To Be Equal