I posted this question on mathstack a couple of weeks ago and even with 100 bounty on it Ive not been able to get any feedback. Hence I tought Id try posting it here.

Ive been told and been reading in some textbooks on SDE's that an SDE or stochastic differential really is an integral equation. In other words, that

$ dX= \beta dt + \sigma dW$ $\,$ "really means" $\,$ $X_{t}= X_{0} +\int_{0}^{t} \beta dt +\int_{0}^{t} \sigma dW$

However I am getting the impression that this is not the case and that $dX$ really is a measure type object which we in turn can "take the integral off", giving us the latter above. This in turn implies that the SDE really has a meaning maybe not as a differental equation in the usual sense but as something else. This impression was obtained for instance from comments in the following post,

In particular the discussion involving $dX=dY$, $AdX=AdY$ and then taking the integral of that.

With that background I ask the following;

Is an SDE then really equal to an integral equation? This would be the same as saying or declaring rather that $\mu= d\mu$, which is fine if one sticks to that convention, but that dosnt seam to be the case in some calculations with differentials.

I also wonder if it is possible to make sense of the $dX$ if one uses more advanced tools? Similar to the case of $dx$ in a ODE, that indeed justify some "informal" operations. My impression is that in general the $dx$ can be tought of as a measure, a differential form etc..

Found these;

https://math.stackexchange.com/questions/2850431/are-sdes-really-differential?rq=1

https://math.stackexchange.com/questions/930578/problem-with-understading-mixed-integration?rq=1,

that looks like similar questions, but which are not answering my question.

Update

While reading Revus and Yors book Continuous Martingales and Brownian Motion I noticed that they denote the equation $e_{x}(f,g)$ for drift and diffusion coefficents $f$ and $g$.

It thus looks like they deliberately avoid what seem to be convention in other books to give the meaning to the differential version to be equal to the integral version.

I suppose the reason for this is to not get into the whole mess that I got into, but that just a guess on my part.

Any thoughts on this would also be great to hear.

Update 2

Comments on why this question is so hard to answer, which seams to be the case ,would also be great. After all it has 8 upvotes and has not been closed due to being regarded as nonsense. Someone with experience must have tought about this before since it is a rather elementary matter.