In calculus, if $\frac{dy}{dx}$ is not in fact a fraction, is the equation below for geometric brownian motion technically incorrect?

Question

Consider the following stochastic differential equation for geometric Brownian motion: $$ {\displaystyle dS_{t}=\mu S_{t}\,dt+\sigma S_{t}\,dW_{t}} $$ I was reading on Wikipedia about geometric Brownian motion (https://en.wikipedia.org/wiki/Geometric_Brownian_motion) and this question popped into my mind. To be clear, I understand the interpretation of the above equation, but is it technically correct? Clearly, if they weren't talking about derivatives (that is not in the limit as $dt$ goes to $0$), then I'd see no problem; but it looks like they simply multiplied both sides of the "proper equation" by $dt$. By the "proper equation" I mean this: $$ {\displaystyle \frac{dS_{t}}{dt}=\mu S_{t}+\sigma S_{t}\,\frac{dW_{t}}{dt}} $$ Am I missing something?

Answer 1

As suggested in the comments, the paths of $W_t$ (and also $S_t$) are almost surely nowhere differentiable, so the notation $\frac{dW_t}{d t}$ is meaningless in this context. The SDE as written is simply notational sugar for:

$$S_t = S_0 + \int_0^t \mu S_r dr + \int_0^t \sigma S_r dW_r$$