Solving SDE's on subsets of $R^n$.

Question

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).

Answer 1

I think the crucial question is how you (or the authors you refer to) interpret the word "Lipschitz". If $b$ and $\sigma$ are ${\it \mbox{globally}}$ Lipschitz, you may indeed extend them by your favorite method to the whole space. Of course, the solutions may differ, if you use different Lipschitz extensions. However, they agree up to the first exit time of $D$, or equivalently, the processes stopped at $\tau_D$ agree (and I assume that is everything what the authors care about).

If $b$ and $\sigma$ are merely ${\it \mbox{locally}}$ Lipschitz, a Lipschitz continuation may not exist. If you have merely continuous coefficients, you may prove local existence of solutions, but these solutions may only exist locally (up to an explosion time). I think the usual way is to make a one-point compactification of $D$ by adjoining an additional 'cemetery' state $\Delta$ to $D$ and look on the SDE on $D \cup \{\Delta\}$.

A classical reference on this is Chapter III of the book "Stochastic Differential Equations and Diffusion Processes" by Ikeda and Watanabe.

Answer 2

This is more of a suggestion than an answer, but I came across a similar issue with regard to Itō's formula (i.e., a C^2 function f is defined only on some domain, and you want to show that Itō's formula holds until the process exits that domain). After lengthy discussions with my professor, I decided that the best way to prove this is by modifying the proof of the usual result. Along the lines of your suggestions, I first tried to use the result on R^n somehow. In the end I found such approaches to be unsatisfactory. On the other hand, I found working through the proof of the result on R^n and appropriately generalizing lemmas, etc., to be enlightening. My professor agreed that this is how I should think about it.

I haven't done this with the SDE uniqueness theorem, but the situation seems analogous. Hopefully it works out similarly.

Answer 3

One idea is to approach the problem through stochastic differential equations with reflections off the boundary of domains. For example, this paper by Tanaka (1979) considers a stochastic differential equation with reflections off the boundary of a convex set. They prove existence and uniqueness for coefficients defined on the domain only, and as far as I can see, in the proof they don't extend the coefficients to all of $\mathbb{R}^d$. If unique solutions exist with reflecting boundary conditions for all time, then presumably the original equation has a solution up to hitting the boundary. A more recent reference is Lions and Sznitman (1984).

This is a bit inelegant in that an SDE stopped at the boundary should be simpler than an SDE reflected off a boundary. But a least you don't need to extend the domain of the coefficients.