5 — Filtrations
Let $(\Omega, \mathcal F, P)$ be a probability space.
Filtration
A filtration is a sequence of $\sigma$-fields \(\{\mathcal F_n\}_{n\ge 1}, \qquad \mathcal F_n \subseteq \mathcal F_{n+1}.\)
Stopping Time
A random time $T:\Omega\to \mathbb Z^+$ is a stopping time if \(\{T=n\}\in \mathcal F_n,\qquad n\ge 1.\)
Sigma-field at a Stopping Time
\(\mathcal F_T = \{A\in\mathcal F : A\cap\{T=n\} \in\mathcal F_n \text{ for all } n\ge 1\}.\)
Natural Filtration
Given a stochastic process ${X_n}_{n\ge 1}$: \(\mathcal F_n = \sigma(X_1,\dots,X_n),\qquad n\ge 1.\)
Then \(\mathcal F_T = \sigma(X_{n\wedge T}).\)
Monotone Stopping Times
If $T_1 < T_2$ almost surely and both are stopping times, then \(\mathcal F_{T_1} \subseteq \mathcal F_{T_2}.\)
Corollary.
If $T_1 < T_2 < \dots$ is an increasing sequence of stopping times, then
${\mathcal F_{T_k}}_{k\ge 1}$ is a filtration.
Theorem
Let ${X_n}_{n\ge 1}$ be IID.
Let $T$ be a stopping time with respect to the natural filtration and assume
\(P(T<\infty)=1.\)
Then:
- \[\{X_{T+n}\}_{n\ge 1} \stackrel{D}{=} \{X_n\}_{n\ge 1}.\]
- ${X_{T+n}}_{n\ge 1}$ is IID and independent of $\mathcal F_T$.
Example: Simple Symmetric Random Walk
\(S_0 = 0,\qquad S_n = \sum_{k=1}^n X_k,\qquad X_k=\pm 1 \text{ with probability } \tfrac12.\)
Stopping times:
- \[T_1 = \min\{n>0 : S_n = 0\}.\]
- \[T_i = \min\{n > T_{i-1} : S_n = 0\}.\]
Then $T_i - T_{i-1}$ are IID.
Wald’s First Equation
Let $S_n = \sum_{k=1}^n X_k$ and $T$ a stopping time with $E[T]<\infty$ and $E|X_1|<\infty$.
Then: \(E(S_T)=E(X_1)E(T).\)
Sketch: \(S_T = \sum_{n\ge 0} (S_{T\wedge(n+1)} - S_{T\wedge n}),\) and \(S_{T\wedge(n+1)} - S_{T\wedge n} = X_{n+1}1_{\{T\ge n+1\}}.\) Since the indicator is $\mathcal F_n$-measurable and $X_{n+1}$ is independent: \(E(S_{T\wedge(n+1)} - S_{T\wedge n}) = E(X_1)P(T\ge n+1).\)
Summing gives Wald’s equation.
Generalization
If $X_{n+1}$ is independent of a smaller filtration $\mathcal G_n$ and $T$ is a stopping time with respect to $\mathcal G_n$, Wald’s equation still holds.
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