# A hitting time of an open set for a càdlàg process is a stopping time

In Protter Stochastic Integration and Differential Equations, Springer (2003), the following definition is given:

Definition. Let $$X$$ be a stochastic process and let $$\Delta$$ be a Borel set in $$\mathbb{R}$$. Define $$T(\omega) = \inf \{t > 0 : X_t \in \Delta \},$$ Then $$T$$ is a hitting time of $$\Delta$$ for $$X$$.

Then the following theorem is stated:

Theorem Let $$X$$ be an adapted càdlàg stochastic process, and let $$\Delta$$ be an open set. Then the hitting time of $$\Delta$$ is a stopping time.

In the proof, Protter states that it is sufficient to show that $$\{T < t\} \in \mathcal{F}_t$$ for $$0 \leq t < \infty$$ (under the condition that $$\mathcal{F}_t$$ is right-continuous).

However, he then states that:

$$\{T < t\} = \bigcup_{s \in \mathbb{Q}\cap[0,t)} \{X_s \in \Delta\}$$ "since $$\Delta$$ is an open set and $$X$$ has right continuous paths". I do not manage to understand this last equality, even with that explanation.

Could someone help me to understand it?

Thanks

• You just need to show that both sides are mutually subset of each other, given the density of $\mathbb{Q}$ in $\mathbb{R}$. – Gordon Dec 17 '18 at 17:53