Let $A\subset\mathbb{R}^n$ be a compact set. Suppose I have an SDE in $\mathbb{R}^n$ of the form $$d X_t = b(X_t)dt + \sigma(X_t)dW_t$$ with $X_0=x_0\in A$, where $b$ and $\sigma$ are continuous but not necessarily bounded.

I am having a hard time formulating the existence of a solution taking values in $A$. Intuitively, I would want to stop the process $X$ at $\tau = \inf\left\{t\geq 0 : X_t\notin A\right\}$. Then, $b$ and $\sigma$ are bounded over $A$ so Theorem 5.4.22 in Karatzas and Shreve gives us existence. Is it rigorous to proceed in this way? If $X$ doesn't necessarily exist, how can we define $\tau$?



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