I posted this on mathstackexchange to no avail.

It is well-known (see for instance Oskendal's text) that if $T>0$ and $$b(\cdot,\cdot): [0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}^n~~~~~~\sigma(\cdot,\cdot):[0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}^{n \times m}$$ are measurable functions and for some $C>0$ we have $$|b(t,x)|+|\sigma(t,x)| \leq C(1+|x|)$$ and $$|b(t,x)-b(t,y)|+|\sigma(t,x)-\sigma(t,y)| \leq C|x-y|,$$ then there is a unique strong solution to the SDE $$dX_t=b(t,X_t)dt+\sigma(t,X_t)dB_t.$$

I have seen papers/books talk about solving this SDE for $t<\tau_D$ where $\tau_D$ is the first time $X_t$ leaves an open set $D$. In this case, we may only have $b$ and $\sigma$ defined on $[0,T] \times D$. I am wondering what precisely is meant by this.

One interpretation is to extend $b$ and $\sigma$ to functions defined on $[0,T] \times \mathbb{R}^n$ by defining $b(t,x)=\sigma(t,x)=0$ for $x \notin D$, but if we do this we kill the Lipschitz property and no longer are guaranteed uniqueness (intuitively it seems like what happens outside of $D$ should not effect what happens to the process before it leaves $D$, but . . .). Another way is to use something like Kirszbraun's Theorem to extend $b$ and $\sigma$ to Lipschitz functions functions on $[0,T] \times \mathbb{R}^n$, use the existence theorem to find a solution to the SDE and then stop the solution at $\tau_D$. Since there might be multiple ways to extend $b$ and $\sigma$, it's not immediately clear that this solution is unique. I think you can argue if it weren't (at least when $b$, $\sigma$ don't depend on $t$), you could piece together a solution up to time $\tau_D$ and a solution from $\tau_D$ to $T$ to contradict the uniqueness of the solution up until time $T$. This all seems too complicated though.

I am wondering what the "right" way of thinking about this sort of thing is (a reference to a book would be great).

**EDIT**

A useful text that was suggested to me on math.stackexchange is this. It appears the right way to think about things is indeed to take a global Lipschitz extension and then solve the resulting equation (and argue that up until $\tau_D$ the solution does not depend on the extension).