8 Debut Theorem

Theorem 8.1

If \(X : T \to \Omega \to E\) is a progressively measurable process with respect to a right-continuous filtration and \(B\) is a Borel-measurable subset of \(E\), then the hitting time of \(X\) in \(B\) is a stopping time.

Proof ▶

8.1 Corollaries

TODO: need to generalize the leastGE definition to general index sets that are not necessarily \(\mathbb {N}\).

Definition 8.2

✓

For a process \(X : ι \to Ω \to ℝ\) and a real number \(a\), define the random time

\begin{align*} \tau _{X \ge a} = \inf \{ t \in ι \mid X_t \ge a\} \: , \end{align*}

in which the infimum is infinite if the set is empty.

Lemma 8.3

If \(X : ι \to Ω \to ℝ\) is a progressively measurable process with respect to a right-continuous filtration, then for any \(a \in \mathbb {R}\), the random time \(\tau _{X \ge a}\) is a stopping time.

Proof ▶

This is a direct application of Theorem 8.1 with the set \(B = [a, +\infty )\).

Corollary 8.4

If \(X : ι \to Ω \to ℝ\) is a right-continuous and adapted process with respect to a right-continuous filtration, then for any \(a \in \mathbb {R}\), the random time \(\tau _{X \ge a}\) is a stopping time.

Proof ▶

This follows from Lemma 8.3 since \(X\) is progressively measurable by Lemma 7.4.