Stopping Time

Definition Stopping Time

• \((\Omega, \mathcal{F}, P, \{\mathcal{F}_{t}\}_{0 \le t})\),
  • probablility space with a filtration
• $T: \Omega \rightarrow [0, \infty)$,
  • r.v.

$T$ is said to be a stopping time of the filtration if

\[\{T \le t\} \in \mathcal{F}_{t}\]

for every $t \in [0, \infty)$.

$T$ is said to be a optional time of the filtration if

\[\{T < t\} \in \mathcal{F}_{t}\]

for every $t \in [0, \infty)$.

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Definition hitting time

• $X: [0, \infty) \times \omega \rightarrow \mathbb{R}$
  • stochastic process
• $A \subseteq \mathbb{R}$,

A stopping time

\[\tau_{A}(\omega) := \inf \{ t \in T \mid X_{t}(\omega) \in A \}\]

is called a hitting time.

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Lemmna

• $T, S: \Omega \rightarrow [0, \infty)$,
  • stopping time

The followings are also stopping time;

• $A := \min(T, S)$,
• $B := \max(T, S)$,
• $C := T + S$,

proof

\[\begin{eqnarray} \{\min(T, S) \le t\} & = & \{T \le t\} \cup \{S \le t\} \nonumber \\ \{\max(T, S) \le t\} & = & \{T \le t\} \cap \{S \le t\} \nonumber \end{eqnarray}\]

All of sets are $\mathcal{F}_{t}$ measurable sets.

$\Box$

Definition

• $T$,
  • stopping time of the filtration \(\{\mathcal{F}_{t}\}\),

The $\sigma$-field $\mathcal{F}_{T}$ of events determined prior to the stotping time $T$ is defined

\[\mathcal{F}_{T} := \{ A \in \mathcal{F} \mid A \cap \{T \le t\} \in \mathcal{F}_{t}, \ t \ge 0 \} .\]

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Theorem Wald’s equation

• $\tau$,
  • discrete stopping time
• $X_{n}$
  • i.i.d. discrete stochastic process with expectation $E[X_{n}]$.

If $E[\tau] < \infty$ and $E[\abs{X}] < \infty$,

\[\mathrm{E} \left[ \sum_{n=1}^{\tau} X_{n} \right] = E[\tau] E[X]\]

proof

$\Box$

Reference

• https://www.columbia.edu/~ks20/stochastic-I/stochastic-I-ST.pdf