In the 5th page of Black and Scholes' original paper on option pricing formulas, they write the following assumption:
b) The stock price follows a random walk in continuous time with a variance rate proportional to the square of the stock price. Thus the distribution of possible stock prices at the end of any finite interval is log- normal. The variance rate of the return on the stock is constant.
I'm a bit confused about the first sentence and how it relates to the GBM assumption $$dS_t = S_t[ \mu dt + \sigma dW_t],$$ especially since the last sentence says that the "variance rate of return" (which I assume is $\sigma^2$) is constant over time. In particular, I don't see where $S_t^2$ comes into play.
Also, I read somewhere that Merton cleaned up the derivation and put it in more mathematically rigorous language. Where can I find that paper?
Answer: if $dX_t = \mu(t,X_t,\dots) dt + \sigma(t,X_t, \dots) dW_t$, then $\mu$ is called the "drift rate" and $\sigma^2$ is called the variance rate. The connection to quadratic variation is that $d\langle X \rangle_t = \sigma(t,X_t,\dots)^2 dt$, i.e. the drift rate of the quadratic variation of $X_t$ is in fact the variance rate of $X_t$.