Brownian motion : what is it exactly and why is it so important? 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 ?
 A: Brownian motion dipicts diffusion, so assuming you have a graph and you are at some node, you want to run simulation where you would end up after some time by just moving to some "immediate node" randomly, now for a brownian motion you cant simply hop to some node that is not connected to you directly, this is an important aspect of brownian motion that it prevents you to go to places for which you have no direct connection. Now you can think of many real life examples where this is relavant. One example is if you want to perform simulations such as Financial stock market where you simply simulate return of your investment, now if you "invest in a stock randomly" you are taking a random walk and the outcome will reflect how much is your return. You can run multiple simulations and determine on average how much money you will make if you invest in companies using random walk "it would be weird that you start from \$0.5 and end up \$1million in a single hop" :)
Now this should give you intuition of scenarios where we can use brownian motion to mimic real scenarios.
