question with stopping times

Question

Suppose we define for $A\in \mathcal{B}(\mathbb{R^n})$ the first hitting time

$$ T_A:= \inf\{t\ge 0;X_t(\omega)\in A\}$$

where $X=(X_t)$ is a stochastic process, adapted to a Filtration and with right-continuous paths. Now suppose that $A$ is open then I want to show:

$$\{T_A<t\}\in \mathcal{F}_t$$

for all $t\ge0$ and where $\mathcal{F}_t$ is a element of the filtration. In the book there's a hint, we should show $\{T_A<t\}=\cup_{q\in \mathbb{Q},0\le q<t}\{X_q\in A\}$. One inclusion, i.e $\supset$,is obvious. Unfortunately I got stuck at the other. Since the paths are right continuous, my first thought was to construct a sequence of rational decreasing to $T_A$, however I do not see why such a sequence should exist and I do not see how I should use that $A$ is open. Some help would be appreciated!

hulik

Answer 1

Hint: Consider the following: $$\{T_A<t\}=\cup_{0\le s<t}\{X_s\in A\}=\cup_{q\in \mathbb{Q},0\le q<t}\{X_q\in A\}.$$ The first equation is easy, but the second equation is troubling you. You should try to understand why if $X_s\in A$ for some real value $s\in[0,t)$, then also $X_q\in A$ for some rational $q\in[0,t)$. This is where you use the fact that $A$ is open and that the sample path is right-continuous.