Checking a solution to the linear homogeneous SDE

Question

This is an excerpt from Mikosch's Elementary Stochastic Calculus (Chapter 3.3):

Here is my question:

What does checking $X$ being a solution to (3.38) mean? (As I understand, one uses the transformation $Y_t=\ln X_t$ and applies Ito's lemma to give the solution (3.39) so that one has (3.40), which is supposed to be the solution.)

Answer 1

The reason why you have to check that $X_t$ is indeed a solution to the SDE is the following step when deriving the candidate for the solution:

Since we expect an exponential form of the solution, we may assume $X_t>0$. This allows us to consider $Y_t = \ln X_t$ [...]

Under the assumption that the solution is strictly positive, we can apply Itô's formula to $\ln(X_t)$.

This is a very typical thing when you try to find the solution to SDE - first you assume that everything is nice and apply Itô's formula (even if you don't know whether you are allowed to do so); as soon as you have found a candidate for the solution you have to verify that this candidate is indeed a solution. Usually the easiest way is to apply Itô's formula; that's a straight-forward calculation.