What do the "d" in SDE notation mean?

Question

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.

Answer 1

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.