What do the "d" in SDE notation mean? An SDE is often written in the form $ dX_t=\mu dt + \sigma dW_t $.
What is the meaning of this equation in English?
If I had to construct an SDE, I would write something like $ \frac{dX_t}{dt} = \mu + \sigma \frac{dW_t}{dt} $. Why are SDEs not written in that way?
I know that the Brownian motion does not have a derivative so it has to do something with that fact but I don't get the real meaning of the standard notation.

An SDE is just a short way of writing the stochastic integral equation 
$$X_t = X_0 + \int_0^t \mu ds + \int_0^t \sigma dW_s$$
So if we take the SDE form and integrate both sides we get:
$$\int_0^t dX_s=\int_0^t\mu ds + \int_0^t\sigma dW_s$$
With the natural equation $$\int_0^t dX_s = X_t - X_0$$ we get original stochastic integral equation.
So the SDE is nothing else then another way of writing the integral equation.
