equities - What are the units of the variables appearing in a standard stochastic differential equation for a Wiener process?

The Black Scholes model assumes the following form for the Wiener process describing the evolution of the stock price S:

$dS=\mu S dt + \sigma S dX$

Clearly $S$ and $dt$ have units of dollars (say) and days (say), respectively. That means $\mu$ has units of "per day". What are the units of the other variables: $\sigma$ and $dX$ ?

At no point in my textbook or any other derivation I've seen is a normalisation performed, so I assume these variables retain some meaningful units. I can't find a textbook that mentions the units, and would like to set the record straight.

Answer

$ \sigma S $ is in units of dollars per square root of a unit of time.

$ \sigma $ is usually quoted as an annual or daily percentage.

$ dX ^2 $ is in units of time, as $ E[(dX)^2] = dt $.

Here is an online tutorial which you may find helpful.

EDIT by kotozna: $\sigma$ has dimensions 1/(square root of time) and $dX$ has dimensions square root of time. Note that $\sigma$ corresponds to but is not exactly the standard deviation.