I am trying to understand and develop some intuition for Brownian motion. As I understand it is characterized by four properties.

$1) W_0 = 0 $

$2) W_t $ has independent increments

3) The increments $ W_{t+u}-W_t$ are normally distributed with mean 0 and variance $u$

4) $W_t$ is continuous

What I don't understand is 3). Why is it natural to assume that the variance is given by u? Is this something that is fundamental to stochastic processes in nature? Why would some other scaling with u not be just as natural a choice?

My second question is about, how I can simulate Brownian motion. I can imagine initiating a stochastic process with events separated by time increments of dt. In this case I could then require that $W_i - W_{i-1}$ is normally distributed with variance dt. But is this enough to guarantee the general condition 3)? For example, how do I know that $W_i - W_{i-2}$ is normally distributed with variance 2dt?



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