Filtrations, adapted processes and stopping times

The lesson "Probability spaces, σ-algebras and random variables" modelled a state of information as a σ-algebra and showed that, on a finite sample space, a σ-algebra is a partition into atoms. Prices are observed over time, so information grows. This lesson makes that growth precise with filtrations, defines which processes are known at each time (adapted) or one step ahead (predictable), and defines stopping times, the random times at which one can act without seeing the future. Three tosses of a fair coin serve as the running example.

Filtrations

Fix a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and an index set of times $I$, either $\{0, 1, 2, \ldots\}$ or $[0, \infty)$. A filtration is a family $(\mathcal{F}_t)_{t \in I}$ of σ-algebras with

$\mathcal{F}_t$ is the collection of events whose occurrence is decided by time $t$. The inclusion says that information is never lost: an event that is decided at time $s$ is still decided at any later time. The quadruple $(\Omega, \mathcal{F}, (\mathcal{F}_t), \mathbb{P})$ is a filtered probability space.

Given a process $(X_t)_{t \in I}$, its natural filtration is the information generated by its own history,

the smallest σ-algebra for which every $X_s$ with $s \le t$ is measurable. A filtration may be larger than the natural one, for example when an investor also watches other assets, but never smaller if the investor sees $X$.

Three coin flips

Toss a fair coin three times: $\Omega = \{H, T\}^3$ has 8 outcomes, each with probability $\tfrac18$, and $\mathcal{F} = 2^\Omega$. Let $\xi_k(\omega)$ be the $k$-th toss and $\mathcal{F}_k = \sigma(\xi_1, \ldots, \xi_k)$, with $\mathcal{F}_0 = \{\emptyset, \Omega\}$. By the previous lesson each $\mathcal{F}_k$ is a partition. Its atoms are the sets of outcomes that agree on the first $k$ tosses: there are $2^k$ of them, each containing $2^{3-k}$ outcomes. The number of events is therefore

Each atom of $\mathcal{F}_k$ splits into two atoms of $\mathcal{F}_{k+1}$, which is the inclusion $\mathcal{F}_k \subseteq \mathcal{F}_{k+1}$ seen as a refinement of partitions. A fair picture is a binary tree: a node at depth $k$ is an atom of $\mathcal{F}_k$, and the leaves below it are the outcomes it contains.

The figure draws the 8 paths as such a tree. Step the time $k$ from 0 to 3 with the arrows or press play: the nodes up to depth $k$ appear and the leaves are grouped into the $2^k$ atoms of $\mathcal{F}_k$. At $k = 0$ all 8 leaves form one block; at $k = 3$ every leaf is its own block.

Adapted processes

A process $(X_t)$ is adapted to $(\mathcal{F}_t)$ if $X_t$ is $\mathcal{F}_t$-measurable for every $t$: its current value is known from the current information. On the coin space, the Doob–Dynkin lemma from the previous lesson says that $X_k$ is adapted exactly when $X_k = f_k(\xi_1, \ldots, \xi_k)$ for some function $f_k$. The number of heads so far, $S_k = \sum_{j \le k} \mathbf{1}\{\xi_j = H\}$, and the walk $W_k = \sum_{j \le k} (2\cdot\mathbf{1}\{\xi_j = H\} - 1)$ are adapted. The process $Y_k = S_3$ for all $k$, which reveals the final number of heads at time 0, is not: $Y_0$ is not constant on the single atom $\Omega$ of $\mathcal{F}_0$. Every process is adapted to its own natural filtration.

Predictable processes

A discrete-time process $(H_k)_{k \ge 1}$ is predictable if $H_k$ is $\mathcal{F}_{k-1}$-measurable for every $k \ge 1$: it is known one step before the time it is indexed by. This is the shape of a trading strategy. $H_k$ is the number of units held over the period from $k-1$ to $k$, chosen at time $k-1$, so the gain from holding a walk $W$ is

Predictability makes such games fair. The increment $W_k - W_{k-1}$ is independent of $\mathcal{F}_{k-1}$ and has mean 0, while $H_k$ is $\mathcal{F}_{k-1}$-measurable, so for bounded $H$ each term has $\mathbb{E}[H_k(W_k - W_{k-1})] = \mathbb{E}[H_k]\cdot 0 = 0$ and $\mathbb{E}[G_n] = 0$. The strategy "hold one unit after a tail, none after a head" is predictable and earns 0 on average. The strategy $H_k = \mathbf{1}\{\xi_k = H\}$ uses the $k$-th toss to decide the $k$-th position. It is adapted but not predictable, and it gains 1 on every head, so $\mathbb{E}[G_3] = 1.5$: it is a model of trading on information from the future.

Stopping times

A stopping time is a random time $\tau : \Omega \to I \cup \{\infty\}$ such that

Whether $\tau$ has already happened by time $t$ must be decided by the information at time $t$. The value $\tau = \infty$ means the time never occurs. In discrete time the condition is equivalent to $\{\tau = k\} \in \mathcal{F}_k$ for every $k$. Proof: if all $\{\tau \le k\} \in \mathcal{F}_k$, then $\{\tau = k\} = \{\tau \le k\} \setminus \{\tau \le k-1\}$ is in $\mathcal{F}_k$ because $\mathcal{F}_{k-1} \subseteq \mathcal{F}_k$. Conversely, $\{\tau \le k\} = \bigcup_{j \le k} \{\tau = j\}$, a union of events from $\mathcal{F}_j \subseteq \mathcal{F}_k$.

A stopping time is a rule for when to act that uses only the past and present: $\{\tau \le t\} \in \mathcal{F}_t$ for all $t$. Exercise dates of American options and stop-loss triggers are stopping times; "sell at the highest price of the year" is not.

Simple facts follow directly from the definition. A constant $\tau = c$ is a stopping time, since $\{\tau \le t\}$ is $\emptyset$ or $\Omega$. If $\tau$ and $\rho$ are stopping times, so are $\min(\tau, \rho)$ and $\max(\tau, \rho)$, because $\{\min(\tau,\rho) \le t\} = \{\tau \le t\} \cup \{\rho \le t\}$ and $\{\max(\tau,\rho) \le t\} = \{\tau \le t\} \cap \{\rho \le t\}$.

First hitting times

Let $X$ be adapted and $A$ a Borel set. The first hitting time of $A$ in discrete time is $\tau_A = \min\{k \ge 0 : X_k \in A\}$, with $\tau_A = \infty$ if $X$ never enters $A$. It is a stopping time because

each $\{X_j \in A\}$ being in $\mathcal{F}_j \subseteq \mathcal{F}_k$. On the coin space take $\tau$ = the time of the first head. Then $\{\tau \le 1\}$ is the atom "first toss H", $\{\tau \le 2\}$ is the union of the atoms HH, HT and TH of $\mathcal{F}_2$, and $\{\tau \le 3\}$ is everything except TTT. Its law is $\mathbb{P}(\tau = k) = 2^{-k}$ for $k = 1, 2, 3$ and $\mathbb{P}(\tau = \infty) = \tfrac18$, so $\mathbb{P}(\tau \le 2) = \tfrac12 + \tfrac14 = 0.75$.

Last exit times

Now let $\sigma$ be the time of the last head among the three tosses, with no value if all three are tails. It is not a stopping time. At $k = 1$, $\{\sigma \le 1\} = \{HTT\}$, which is not a union of atoms of $\mathcal{F}_1$: the outcomes HTT and HTH agree on the first toss, yet $\sigma(HTT) = 1 \le 1$ and $\sigma(HTH) = 3 > 1$. At $k = 2$, $\{\sigma \le 2\} = \{HHT, HTT, THT\}$ splits the atoms HH, HT and TH of $\mathcal{F}_2$. To know at time 1 that the last head has already happened, you would need to see tosses 2 and 3.

The figure lets you compare the two rules. Switch the random time between "first H" and "last H". Each leaf shows the time for its path, or "none". For the current $k$ the readout says whether $\{\tau \le k\}$ is a union of atoms of $\mathcal{F}_k$, and for "last H" it shows a pair of paths that agree up to time $k$ but disagree on whether $\tau \le k$. Step through $k$ and note that "first H" passes at every $k$, while "last H" fails at $k = 1$ and $k = 2$.

The σ-algebra at a stopping time

The information available at a stopping time $\tau$ is the σ-algebra

the events whose occurrence is known once $\tau$ has happened. For $\tau$ = first head, the event "toss 2 was H" belongs to $\mathcal{F}_\tau$ only together with the knowledge of $\tau$: on $\{\tau = 1\}$ it is not yet decided, on $\{\tau = 2\}$ it is. The strong Markov property and optional stopping, in later lessons, are statements about $\mathcal{F}_\tau$.

Brownian motion in continuous time

For Brownian motion $W$ the natural filtration is $\mathcal{F}^W_t = \sigma(W_s : s \le t)$. $W$ is adapted to it, and by independent increments every future increment $W_u - W_t$, $u > t$, is independent of $\mathcal{F}^W_t$. The first passage time $\tau_a = \inf\{t \ge 0 : W_t = a\}$ to a level $a > 0$ is a stopping time. Since paths are continuous and start at $0 < a$,

which uses only countably many times $s \le t$ and so lies in $\mathcal{F}^W_t$. For hitting times of open sets, and for many other constructions, one needs a slightly enlarged filtration satisfying the usual conditions: it is right-continuous, $\mathcal{F}_t = \bigcap_{s > t} \mathcal{F}_s$, and $\mathcal{F}_0$ contains all $\mathbb{P}$-null sets. Later lessons assume these without further comment.

Summary

• A filtration $(\mathcal{F}_t)$ is an increasing family of σ-algebras; $\mathcal{F}_t$ is the information at time $t$, and the natural filtration $\sigma(X_s : s \le t)$ is the history of $X$.
• With $n$ coin tosses, $\mathcal{F}_k$ is the partition into $2^k$ atoms of outcomes that agree on the first $k$ tosses, with $2^{2^k}$ events.
• Adapted means $X_t$ is $\mathcal{F}_t$-measurable; predictable means $H_k$ is $\mathcal{F}_{k-1}$-measurable, the form of a trading position chosen in advance.
• A stopping time satisfies $\{\tau \le t\} \in \mathcal{F}_t$. First hitting times are stopping times; last exit times are not.
• For Brownian motion, $\mathcal{F}^W_t$ is independent of future increments, and first passage times are stopping times.