The Black Scholes model assumes the following form for the Wiener process describing the evolution of the stock price
$dS=\mu S dt + \sigma S dX$
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.