Filtration

Definition 5.3: (Filtration). Given a probability space, the filtration is a nondecreaseing family of $\sigma$-algebras $\{\mathcal{F}_t\}_{t \leq 0}$ such that $\mathcal{F}_s \subset \mathcal{F}_t \subset \mathcal{F}$ for all $0 \leq s < t$.

Intuitively, the filtration is a sigma algebra of events that can be determined before time $t$ (we can’t lose information by foing forward in time). A stochastic process is called $\mathcal{F}_t$-adapted if it is measurable with respect to $\mathcal{F}_t$; that is, for all $B \in \mathcal{R}$, $X_t^{-1}(B) \in \mathcal{F}_t$. We can always assume that the $\mathcal{F}_t$ contains $F_t^{X}$ and all sets of measure zero, where $F_t^{X} = \sigma(X_s, s \leq t)$ is the sigma algebra generated by the process $X$ up to time $t$.

As an example, in a series of coin flips, when $n=0$

$$\mathcal{F}_0^X = \{\emptyset, \Omega\}$$

and when $n=1$,

$$\mathcal{F}_1^X = \{\emptyset, \Omega, \{H\}, \{T\}\}$$

when $n=2$,

$$\mathcal{F}_2^X = \sigma(\{\emptyset, \{HH\}, \{TT\}, \{HT\}, \{TH\} \})$$

(I believe this last statement is equivalent to what the book has)

Stopping Time

Definition 5.4: (Stopping time for discrete time stochastic processes). A stopping time is a random variable $T$ taking values in $\{1,2,\ldots\}\cup \{\infty\}$ such that for any $n < \infty$,

$$\{T \leq n\} \in \mathcal{F}_n$$

For the discrete case, it doesn’t matter if we say $\{T \leq n\}$ or $\{T = n\}$ simply becase it has to be satisfied for all $n$.

Proposition 5.5: (Properties of stopping times). For the Markov process $\{X_n\}_{n \in \mathbb{N}}$, we have

  • (1) if $T_1, T_2$ are stopping times, then $T_1 \wedge T_2, T_1 \vee T_2, T_1 + T_2$ are stopping times
  • (2) if $\{T_k\}_{k \geq 1}$ are stopping times then $\sup_k T_k, \inf_k T_k, \limsup_k T_k, \liminf_k T_k$ are stopping times

Definition 5.6: (Stopping time for continuous time stochastic processes). A stopping time is a random variable $T$ taking values in $[0,\infty]$ such that for any $t \in \mathbb{\bar{R}}^+$,

$$\{T \leq t\} \in \mathcal{F}_t$$

Note that we cannot swap the inequality for an equals sign in the definition of a stopping time for continuous time processes. Furthermore, porposition 5.5 holds for conitnious time processes if the filtration is right continuous: $\mathcal{F}_t = \mathcal{F}_{t^+}= \bigcap_{s>t} \mathcal{F}_s$.