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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 $$|b(t,x)|+|\sigma(t,x)| \leq C(1+|x|)~~\mbox{and}~~|b(t,x)-b(t,y)|+|\sigma(t,x)-\sigma(t,y)| \leq C|x-y|,$$ 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).


In addition to the answer below, there is some interesting discussion on this problem here.

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up vote 1 down vote accepted

This is the book you want, the review in the first ~50 pages of SDEs in Euclidean spaces covers this issue.

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This is a useful reference. It seems the approach he takes is what I was thinking, namely to extend the coefficients to be Lipschitz on the entire domain and then solve the resulting SDE (uniquely) and stop your solution the first time you leave $D$. A detail (as far as I can tell) he glosses over is why two different extensions to all of $\mathbf{R}^n$ cannot have different solutions up until $\tau_D$ (I see why this is true when we actually have a diffusion). – ShawnD Mar 15 '12 at 23:12
I think Theorem 1.18 on pages 12-15 specifically addresses the uniqueness issues in depth. – Mr. F Mar 16 '12 at 15:06
Yes, he does. It still feels like some detail is missing, but I think I can fill in the details in the cases I care about. I think I've distilled this down to being able to answer why if there is a unique solution to an $SDE$ up until time $t$ is it necessarily true there is a unique solution up to some stopping time $\tau<t$. – ShawnD Mar 16 '12 at 17:35

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