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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

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

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