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This is a question concerning Remark(i) under Theorem 7.4.1(Dynkin's formula) on Page 124, $\textit{SDE}$, by Oksendal.

It says that if $dX_t=\mu(X_t)dt+\sigma(X_t)dB_t$ is an $n$-dimensional time-homogeneous Ito diffusion, with $B_t$ being $m$-dimensional Brownian motion, and $\tau$ is its first exit time of some bounded set, then $\mathbb{E}(\tau)<\infty$, and then we have some desired properties concerning the theorem. But how to prove $\mathbb{E}(\tau)<\infty$?

My first intuition is using the martingale approach as we do for Brownian motion, like considering $M_t:= (X_t-\int_0^t\mu(X_s)ds)^2-[Y]_t$ with $Y$ being $(X_t-\int_0^t\mu(X_s)ds)$. But this is too implicit and cannot tell me any information about the stopping time $\tau$.

Any help is desired.

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  • $\begingroup$ You'll certainly need some assumptions on $\mu, \sigma$ since it's not true in full generality. (As a trivial example, what if $\mu, \sigma$ vanish on a neighborhood of the boundary of your set? Or what if the drift $\mu$ pushes the process very strongly into the interior?) $\endgroup$
    – Nate Eldredge
    Commented Mar 22, 2021 at 19:49
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    $\begingroup$ A natural approach would be as follows. Let $\phi(x)$ be the probability that $\tau < 1$ when $X_0 = x$. Unless $X_t$ is "degenerate" (in the sense of Nate Eldredge's comment), we have $\phi(x) < 1$ for every $x$. Under some minimal regularity assumptions, $\phi$ should be continuous and vanish on the boundary. Thus, $q:=\|\phi\|_\infty < 1$. Now just use the Markov property to figure out that $\mathbb E(\tau \wedge t) \leqslant 1 + q \mathbb E(\tau \wedge t)$, so that $\mathbb E(\tau \wedge t) \leqslant 1/(1-q)$. By passing to the limit as $t \to \infty$, we obtain $\mathbb E(\tau) < \infty$. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Mar 22, 2021 at 20:31
  • $\begingroup$ @NateEldredge: I think there is only global Lipschitz conditions on them, to assure the uniqueness, at least I didn't find any further requirements given in the book. $\endgroup$
    – MikeG
    Commented Mar 23, 2021 at 5:11
  • $\begingroup$ @MikeG Have you ever found conditions on $\mu$ and $\sigma$ for $\phi(x)<1$ on the interior of the domain? $\endgroup$
    – Math_Newbie
    Commented Mar 5 at 21:07

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