Negative volatility of Ito Diffusion?

Question

This might be a silly question. But I wonder if the volatility or diffusion parameter in Ito diffusion must be positive or not. I.e.

dX=$\mu dt$+$\sigma dz$, where z is a standard brownian motion.

Does $\sigma$ have to be positive? From the perspective of discrete time, negative or positive $\sigma$ both correspond to same variance $\sigma^2$?

Answer 1

Note that $z$ and $-z$ are both brownian motions, so we can always consider it as $dX = \mu dt + (-\sigma) d(-z)$ if we want. It makes no difference whether $\sigma$ is positive or negative (though notationally we typically take $\sigma>0$).