When I look up the definition of 'Wiener process' at Wikipedia, it tells me:
$W(0) = 0$ and $W(t) - W(s) \sim N(0, t-s)$.
When I try to simulate this in matlab, I get different results when I define a vector $W1$ to be like:
$W1 = cumsum(dW)$, where $dW(j) \sim N(0, dt)$,
and a vector $W2$ to be like:
$W2(0) = 0$ and $W2(j) \sim N(0, dt*j)$
$W2$ apparently doesn't look like a Brownian motion, but it is still compliant with the requirements. How comes?