In a probability triple $(\Omega, \mathcal{F}, \mathbb{P})$, a stochastic process $(W_{n})_{n \ge 0}$ is a sequence of random variables.
- The process is **integrable** if all of its RVs are.
> [!definition|*] Filtrations
> A **filtration** is a sequence of nested sub-$\sigma$-algebras $(\mathcal{F}_{n})_{n \ge 0},~ \mathcal{F}_{0} \subseteq \mathcal{F}_{1}\subseteq\dots$. They define the **filtered probability space** $(\Omega, \mathcal{F}, (\mathcal{F}_{n})_{n \ge 0}, \mathbb{P})$.
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- Interpreting $\sigma$-algebra as information, *a filtration represents the growth of knowledge over time (indexed by $n$)*. In particular, we don't forget anything.
- Furthermore, define the "total information" $\mathcal{F}_{\infty}:= \sigma\left(\bigcup_{n=0}^{\infty}\mathcal{F}_{n}\right)$.
A stochastic process $(X_{n})$ is **adapted** to the filtration $(\mathcal{F}_{n})$ if $X_{n} \in m\mathcal{F}_{n}$ for all $n$.
- One particular way of constructing adapted processes is to simply let $\mathcal{F}_{n}:= \sigma(X_{0},\dots,X_{n})$, called the **natural filtration**.
- More generally, if the process $(W_{n})$ is adapted to $(\mathcal{F}_{n})$, then taking $f_{n} \in m \mathcal{F}_{n}$, the process $(X_{n}):=(f_{n}(W_{0},\dots,W_{n}))$ is also adapted to the filtration.
## Stopping Time and Stopped Processes
> [!definition|*] Stopping Times
> A random variable $\tau: \Omega \to \mathbb{N} \cup \{ \infty \}$ is a **stopping time** if $\{ \omega:\tau(\omega)=n \} \in \mathcal{F}_{n}$. That is, *we know whether $\tau$ stops at $n$ with the information available at $n$.*
>
> Note that $\mathbf{1}_{\tau \ge k}=\mathbf{1}_{\tau \not < (k-1)}$ is a predictable process.
> [!examples] Examples of stopping times
> Examples include:
> - The first **hitting time** of a set $A \in \mathcal{F}$, e.g. selling at the first time a stock price $(X_{n})$ shoots over $\$100$ corresponds to the hitting time of $A=[100,\infty)$: $\tau:= \inf \{ n \ge 0~|~X_{n} \in A \}.$
> - Constants $\tau=\tau_{0}$.
>
> Non-examples include:
> - Things that require knowing the future, e.g. date of the warmest day or time of the highest stock price in a year corresponds to $\tau=\underset{0 \le t \le N}{\arg\max} ~X_{t}$ for the temperature / stock price $(X_{t})$.
> - The last hitting time of a set $A \in \mathcal{F}$: $\tau ^{*}:= \sup \{ 0 \le n \le N ~|~ X_{n} \in A\},$since $\{ \tau ^{*}\leq N-1 \} \notin \mathcal{F}_{N-1}$.
> [!definition|*] Stopped Processes
> If $X$ is a stochastic process and $\tau$ a stopping time, both adapted to the filtration $(\mathcal{F}_{n})$, then the **stopped process** $X^\tau$ is $X^{\tau}_{n}:=X_{n \land \tau}, \text{ adapted to } \mathcal{F}_{n \land \tau},$where $n \land \tau:=\min(n, \tau)$. That is, $X$ stops changing after $\tau$.
>
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- $X^{\tau}$ is adapted to $(\mathcal{F}_{n})$ as well.