Integrating both sides of stochastic equation

Question

When I was reading one document, I noticed one integration there, which I found very odd. http://www-2.rotman.utoronto.ca/~hull/TechnicalNotes/TechnicalNote22.pdf I am referring to the part: "Integrating between time 0 and time T". If I understand it correctly, we have a stochastic equation there, for which we take integral. My question is: why can we do that? We have different variables on both sides of equation: dt, dS and d ln S. I was taught that generally we can do such thing, but we should take the integral with respect to the same variable on both sides, e.g. if we have $f(x)=g(y)$, then we can write $\int f(x) dx = \int g(y) dx$. But in above document author just add $\int_{0}^{T}$ to each expression. Why is this correct? Could you please recommend me a good mathematical book which I could read to better understand this concept? Thank you in advance

Answer 1

There is no "concept" behind that step. You just missed the point that an SDE is just a kind of "short" notation e.g. to cite the Wikipedia artical: "This equation should be interpreted as an informal way of expressing the corresponding integral equation."

So if you "integrating between time 0 and time T" it's just another (bad) note that you write the (short) SDE as the corresponding integral equation.

For your question: Every book that has an introduction about SDEs should be useful.