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Let $(\Omega,\mathscr F,(\mathscr F_t)_{t\geq 0},\mathsf P)$ be a complete filtered probability space and $X = (X_t)_{t\geq 0}$ be a cadlag stochastic process with value in a Polish space $E$. Is it true that for any closed subset $A$ of $E$ the first hitting time $ \tau = \inf\{t\geq 0:X_t\in A\} $ is a stopping time, i.e. $\{\tau\leq t\}\in \mathscr F_t$ for all $t\geq 0$; and what if $A$ is an open set.

As an example, we can consider $X$ with values in $\mathbb R$. It holds that the first hitting time of a closed half-line $ \tau = \inf\{t\geq 0:X_t\in[K,\infty)\} $ is a stopping time. Can we apply the following argument:

Let $\tau = \inf\{t\geq 0:X_t\in (K,\infty)\}$ then $ \{\tau\leq t\} = \bigcup\limits_{n=1}^\infty\{\tau_n\leq t\}\in \mathscr F_t $ where $\tau_n = \inf\{t\geq 0:X_t\in[K+1/n,\infty)\}$ and hence $\{\tau_n\leq t\}\in \mathscr F_t$.

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    @TheBridge: thanks for mentioning it, there was a typo. I use this sequence because $ (K,\infty) = \bigcup\limits_{n=1}^\infty[K+1/n,\infty). $ The previous version of course was not correct, fixed it now.2012-02-20

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The measurability of hitting times is a subtle and complex problem, i.e., a pain in the neck. Of course, the raw $\sigma$-fields are pretty much hopeless, even if the process has continuous paths and $A$ is open. (Take two sample paths that start on the boundary of $A$; one stays put while the other immediately heads into $A$. If $\tau_A$ were a stopping time, then $1[\tau_A=0]$ should be a function of the initial position only.)

It is often assumed that the filtration is complete and right continuous, then things work out better. In this case, for open $A$ and $t>0$ we have $\{\tau_A where $Q_t$ is the set of rational numbers in $[0,t)$. This, and the right-continuity of the fields shows that $\tau_A$ is a stopping time.

For general Borel sets, I suggest reading The measurability of hitting times by Richard Bass, Electron. Comm. Probab. 15 (2010) 99-105; 16 (2011) 189-191 for an elementary (but not easy) exposition of quite general results about hitting times. Note that even Prof. Bass made a subtle error, later corrected. The paper and correction are number 130 on his home page:

http://bass.math.uconn.edu/biblio.html

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    By "raw" $\sigma$-fields I mean exactly those ${\cal F}_t$ in your definition. So, no, they aren't right continuous.2017-03-27