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My question is simple : what is it exactly a Brownian motion and why is it so important ?

So, I read the the wiki page of the Brownian motion, and the definition is : continuous stochastic process with independent increments and stationary increment normally distributed. Indeed, I can accept it as a definition, but it doesn't really tell me why such a stochastic process is that important and popular. Because Brownian motion is everywhere in probability, finance... and I really don't get a process with such a definition is so important, so maybe someone can tell me about the motivation behind ?

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    $\begingroup$ You can think of the Brownian motion as the limit of a simple random walk. Random walks are simple but useful models for many processes, and the time-continuous analog is a Brownian motion. $\endgroup$ – Stefan Dec 7 '18 at 10:11
  • $\begingroup$ @Stefan: Thank you. If it's a limit of a simple random walk, why does the brownian motion is constructed as a levy process ? It's not really a simple random, is it ? (because the construction use Fourier series or wavelet)... but both use the same idea of having an orthogonal basis of $L^2(0,1)$ $\endgroup$ – user623855 Dec 7 '18 at 10:40
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    $\begingroup$ See for example en.wikipedia.org/wiki/…, there are many ways to construct Brownian motions, including (but not limited to) fourier / wavelet constructions, or "zooming in" on a random walk. $\endgroup$ – Stefan Dec 7 '18 at 10:44
  • $\begingroup$ It seems to me that the wikipedia article you cite answers your question. $\endgroup$ – zhoraster Dec 7 '18 at 15:44

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