Rational-Time Characterization of Hitting Open Sets by Càdlàg Processes

Summary

The note concerns a measurability step in proving that the first time an adapted càdlàg process enters an open set is a stopping time. The stated argument expresses the event that the hitting time is strictly before a fixed time as a union over rational observation times before that time. Adaptedness makes each rational-time event measurable at its observation time, and hence in the filtration at the fixed time.

The question focuses on why this union captures entry at any real time: openness supplies a neighborhood around the process value that lies in the set, while right continuity keeps the path in that neighborhood for a short interval after entry. A rational time can then be chosen in that interval. This is a mathematical probability concept rather than a trading method. The document states the result under a right-continuous filtration and does not include the answer or explore variants such as closed sets or processes without right-continuous paths.

Key ideas

• The hitting time of an open set for an adapted càdlàg process is presented as a stopping time.
• The event of entry before a fixed time can be tested using rational observation times.
• Openness and right continuity ensure that entry at a real time persists briefly enough to include a rational time.
• The argument relies on the stated filtration assumptions and does not cover other set or path conditions.

Tags

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


# 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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.