Hitting time defined by another hitting time is a stopping time.

Question

I'm trying to prove the next:

Let $(X_{t})_{t\geq 0}$ be a Lévy process. We define $T_{A}=\inf\{t>0:X_{t}\in B\}.$ Also we define $T_{A}^{'}=\inf\{t>T_{A}: X_{t}\in B\}.$ Here $B$ is an open ball centered at $0$. Then $T_{A}^{'}$ is a stopping time.

I know that $T_{A}$ is stopping time when $B\subset\mathbb{R}$ is an open or closed subset, but I'm stuck proving that $T_{A}^{'}$ is stopping time.

I appreciate any kind of help to prove this.

Answer 1

The proof is very similar to the proof that $T_A$ is a stopping time. Note that $T_A'(\omega) \leq t$ for some fixed $\omega \in \Omega$ if, and only if, there exists $q \in \mathbb{Q} \cap (0,t)$ such that the following two conditions are satisfied.

• $T_A(\omega) \leq q$,
• For any $n \in \mathbb{N}$ there exists $r_n \in \mathbb{Q} \cap (q,t+1/n)$ such that $X_{r_n}(\omega) \in B$.

To prove this equivalence you have to use that $B$ is an open set and that $(X_t)_{t \geq 0}$ has right-continuous sample paths. Once you have shown this, you can conclude that

$$\{T_A' \leq t\} = \bigcup_{q \in \mathbb{Q} \cap (0,t)} \left( \{T_A \leq q\} \cap \bigcap_{n \in \mathbb{N}} \bigcup_{r \in \mathbb{Q} \cap (q,t+1/n)} \{X_r \in B\} \right).$$

Since you already know that $T_A$ is a stopping time, this shows that $T_A'$ is a stopping time.