Consider the SDE $${\rm d}X_t=b(X_t){\rm d}t+\sigma(X_t){\rm d}W_t\tag1$$ with Lipschitz continous $b:\mathbb R^d\to\mathbb R^d,\sigma:\mathbb R^{d\times d}\to\mathbb R$ and a $d$-dimensional Brownian motion $(W_t)_{t\ge0}$. Now, let $$\iota:\mathbb R^d\to[0,1)^d\;,\;\;\;x\mapsto x-\lfloor x\rfloor,$$ where $\lfloor x\rfloor:=(\lfloor x_1\rfloor,\ldots,\lfloor x_d\rfloor)$.
Question: Can we prove existence of a solution to an SDE similar to $(1)$, but evolving in $[0,1)^d$ (with toroidal wrap arround at the boundary) instead of $\mathbb R^d$? Would the correct formulation of that SDE be $$X_t=\iota\left(X_0+\int_0^tb(X_s)\:{\rm d}s+\int_0^t\sigma(X_s)\:{\rm d}W_s\right)\tag2?$$
If that's correct, does $(2)$ admit an invariant measure? My hope would be that existence of an invariant measure would be easier to verify here, since one problem in $\mathbb R^d$ is that "all the probability mass could go to infinity"; which is not possible in $[0,1)^d$ with the wrap around.
Can we also give an explicit form of (the density of) that measure?
EDIT: Let me concretize my question: In my actual application, $b,\sigma$ are only defined on $[0,1)^d$. I guess this is not a problem, since we can extend them periodically to $\mathbb R^d$ by $$b(x):=\sum_{k\in\mathbb Z^d}1_{[k,\:k+1)}(x)b(x-k)\tag3$$ and analogously for $\sigma$. Next, I've also got a probability density $p:[0,1)^d\to[0,\infty)$ and would like to see how I need to choose $b$ and $\sigma$ such that the wrapped solution of the SDE has an invariant measure with density $p$.
The paper Langevin diffusions on the torus: estimation and applications is close to what I'm asking, but I don't really understand why Proposition 1 holds (asked for that here) and it is somehow different in the treatment of $p$.