I've been studying stochastic differential equations (SDEs) for about a year now. I'm trying to understand more about uniqueness of solutions and I've noticed something in most of the math that I would like help clarifying. The general form of an SDE is $$dX_t = \mu(t,\omega) dt + \sigma(t, \omega) dW_t$$ But my question is, why in a lot of proofs I see that we just consider SDEs with a constant diffusion function, like this? $$dX_t=\mu(t,\omega) dt + dW_t$$ For example, here is a cited proof of the uniqueness of this type of SDE: https://arxiv.org/pdf/0709.4147.pdf. Also, in this paper where Tzen and Raginsky prove some results on neural SDEs, on the first page they introduce the model, which is an SDE without drift. In fact Raginsky explains in one youtube video how an SDE with Föllmer drift approximates any target measure, and this only needs a drift and identity diffusion term.
And myself I've been pondering this question. I read Steele, Stochastic Calculus and Financial Applications, and there is short section on the existence and uniqueness of solutions of the SDE. Here, they represent the SDE with both drift and diffusion terms. There are restrictions on the Lipshitz condition for the drift and diffusion added together, and also on the variance.
So my question here is why do we often see analysis of SDEs done on these equations with drift but identity diffusion terms? Is there some connection there between the 2 equations? It seems that perhaps we can represent any SDE of the first form (drift and diffusion) with an SDE of the second form (only drift and Brownian motion)?