last updated: October 16, 2024

Stochastic Processes and Financial Mathematics
(part one)

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8.3 The stopped \(\sigma \)-field \(\msconly \)

Given a filtration \((\mc {F}_n)\) and a stopping time \(T\), it is natural to think about the information available up to and including time \(T\). The following definition makes this concept precise.

  • Definition 8.3.1 Let \((\mc {F}_n)\) be a filtration and let \(T\) be a stopping time. Define

    \[\mc {F}_T=\big \{A\in \mc {F}\-A\cap \{T\leq n\}\in \mc {F}_n\text { for all }n\big \}.\]

In words, \(\mc {F}_T\) is often known as the \(\sigma \)-field obtained by stopping \((\mc {F}_n)\) at time \(T\). If \((\mc {F}_n)\) is the generated filtration of \((X_n)\), then \(\mc {F}_T\) holds the information obtained from \((X_1,X_2,\ldots ,X_T)\). We will often need to use our intuition when dealing with this concept, but let us formally prove that \(\mc {F}_T\) is a \(\sigma \)-field.

  • Lemma 8.3.2 Let \((\mc {F}_n)\) be a filtration and let \(T\) be a stopping time. Then \(\mc {F}_T\) is a \(\sigma \)-field,

Proof: We must check the three conditions of Definition 2.1.1. Firstly, for all \(n\in \N \) we have \(\emptyset \cap \{T\leq n\}=\emptyset \in \mc {F}_n\), so \(\emptyset \in \mc {F}_T\). Secondly, if \(A\in \mc {F}_T\) then \(A\cap \{T\leq n\}\in \mc {F}_n\), and \(\{T\leq n\}\in \mc {F}_n\) because \(T\) is a stopping time. Hence

\[(\Omega \sc A)\cap \{T\leq n\}=\{T\leq n\}\sc \l (A\cap \{T\leq n\}\r )\quad \in \mc {F}_n\]

so \(\Omega \sc A\in \mc {F}_T\). Thirdly, if \(A_m\in \mc {F}_T\) for all \(m\), then \(A_m\cap \{T\leq n\}\in \mc {F}_n\) for all \(m,n\). Hence

\[\l (\bigcup _{m=1}^\infty A_m\r )\cap \{T\leq n\}=\bigcup _{m=1}^\infty A_m\cap \{T\leq n\}\quad \in \mc {F}_n,\]

so \(\cup _{m=1}^\infty A_m\in \mc {F}_T\).   ∎