Stopping Times

Motivation: why this matters in quant finance

A stopping time is a random time

\tau

at which you decide to "stop watching" a process — using only information available up to time $\tau$ . No peeking at the future. This is the exact mathematical formalisation of a trading rule: the rule that says "exit when the moving average crosses the price," "close the position if drawdown exceeds 5%," "exercise the American option the moment the continuation value drops below intrinsic." Every rule an algorithmic trader writes is a stopping time, and every theorem about them tells you something about what your rule can and cannot achieve.

Stopping times are also the glue between martingales and actual computations in quant finance. The optional stopping theorem says that fair games stay fair under any reasonable stopping rule — but you cannot even state that theorem without knowing what a stopping time is. First-passage problems — "what is the probability the stock hits

K

before

T

?", "what is the expected time to trigger a VaR breach?", "when does a barrier option knock in?" — are all stopping-time problems, typically solved by martingale methods applied at the stopping time.

More subtly, stopping times appear in the mathematical definition of localisation — the trick used to extend Itô's lemma and stochastic integration from bounded processes to possibly-exploding ones. Local martingales, semimartingales, and BSDE solutions all depend on the ability to stop a process at a well-chosen random time. Without stopping times the whole modern theory of stochastic calculus collapses.

The informal idea

Fix a filtration

(\mathcal{F}_n)

— the information flow through time. A stopping time

\tau

with values in

\{0, 1, 2, \ldots\} \cup \{\infty\}

is a random time such that, at every deterministic time

n

, you can decide from the data available at time

n

whether

\tau

has already occurred. Formally:

\{\tau = n\} \in \mathcal{F}_n \quad\text{for every } n \ge 0.

Equivalent and often more convenient:

\{\tau \le n\} \in \mathcal{F}_n \quad\text{for every } n \ge 0.

Both say the same thing: the event "has the stopping triggered yet?" is $\mathcal{F}_n$ -measurable at every $n$ . You don't need the future to know whether you have stopped.

Examples you already understand:

Deterministic times. $\tau \equiv 5$ is a stopping time: $\{\tau \le n\} = \emptyset$ for $n < 5$ and $= \Omega$ for $n \ge 5$ ; either way the event is in $\mathcal{F}_n$ .
First hitting time. $\tau = \min\{n : S_n \ge 100\}$ for a random walk $S_n$ . At time $n$ , you know the whole path up to $n$ , so you know whether $S_n$ has reached 100 yet.
A non-example. $\tau =$ "the time of the maximum of $S_n$ over $n \in [0, N]$ ." Not a stopping time: at time $n = 5$ you don't know whether $S_5$ is the future maximum; that requires looking ahead.

The "no peeking" rule is what makes stopping times mathematically useful. You cannot engineer a winning strategy by choosing a stopping time that somehow knows the future — the optional stopping theorem makes this precise.

Formal definitions and properties

Discrete-time stopping time

Given a filtration

(\mathcal{F}_n)_{n \ge 0}

, a map

\tau: \Omega \to \{0, 1, 2, \ldots\} \cup \{\infty\}

is a stopping time if

\{\tau \le n\} \in \mathcal{F}_n

for every

n \ge 0

Continuous-time stopping time

For a right-continuous filtration $(\mathcal{F}_t)_{t \ge 0}$ , a map $\tau: \Omega \to [0, \infty]$ is a stopping time if $\{\tau \le t\} \in \mathcal{F}_t$ for every $t \ge 0$ . Equivalently (in the right-continuous case): $\{\tau < t\} \in \mathcal{F}_t$ for every $t$ .

$\sigma$ -algebra at a stopping time

The information available at $\tau$ is:

\mathcal{F}_\tau := \{A \in \mathcal{F} : A \cap \{\tau \le n\} \in \mathcal{F}_n \text{ for every } n\}.

Intuitively, $A \in \mathcal{F}_\tau$ iff you can decide whether $A$ has occurred using only information up to the random time $\tau$ .

Key facts:

$\tau$ is $\mathcal{F}_\tau$ -measurable (you know $\tau$ itself at time $\tau$ ).
If $X_n$ is adapted and $\tau < \infty$ a.s., then $X_\tau$ is $\mathcal{F}_\tau$ -measurable.

Combinators

Stopping times compose nicely. If $\sigma, \tau$ are stopping times, then so are:

$\sigma \wedge \tau := \min(\sigma, \tau)$ — "stop whichever triggers first."
$\sigma \vee \tau := \max(\sigma, \tau)$ — "stop once both have triggered."
$\sigma + c$ for a deterministic constant $c$ — "wait $c$ more steps after $\sigma$ ."

But $\sigma + \tau$ is not necessarily a stopping time — you cannot schedule "stop at

\sigma

plus a value of

\tau

that is revealed later." Always check the filtration structure carefully.

Canonical examples

1. First hitting time of a level

For a random walk

S_n

and a set

B \subseteq \mathbb{R}

, the first hitting time of $B$ is

\tau_B = \inf\{n \ge 0 : S_n \in B\},

with the convention $\inf\emptyset = \infty$ . Standard argument that $\tau_B$ is a stopping time:

\{\tau_B \le n\} = \bigcup_{k=0}^n \{S_k \in B\} \in \mathcal{F}_n,

because each $\{S_k \in B\}$ is in $\mathcal{F}_k \subseteq \mathcal{F}_n$ .

2. First exit time of an interval

For a random walk

S_n

starting at

x \in (a, b)

, the first exit time of $(a, b)$ is

\tau = \min(\tau_a, \tau_b)

, where

\tau_a = \inf\{n : S_n \le a\}

and

\tau_b = \inf\{n : S_n \ge b\}

. As the minimum of two stopping times,

\tau

is a stopping time. This is the gambler's ruin stopping time.

3. American option exercise time

An American option can be exercised at any time $\tau \le T$ chosen by the holder based on available information. The holder's optimisation problem is to maximise $\mathbb{E}^{\mathbb{Q}}[e^{-r\tau}g(S_\tau)]$ over all stopping times $\tau \le T$ , where $g$ is the payoff and $S_t$ is the underlying. The optimal exercise time is a stopping time — the holder cannot foresee the future and must choose using information available at each moment.

4. Drawdown stop

In a trading strategy with wealth $X_n$ , define $\tau = \inf\{n : X_n \le 0.95\cdot \max_{k \le n}X_k\}$ — "stop the moment drawdown exceeds 5%." This is a stopping time: the running maximum and the current value are both in $\mathcal{F}_n$ .

5. A non-stopping time

Last hitting time.

\tau^* = \sup\{n \le N : S_n = 0\}

— "the last time the walk returns to zero before horizon

N

." At

n < N

, you do not know whether the walk will later return to zero, so

\{\tau^* \le n\}

is not in

\mathcal{F}_n

. The last zero is a beautiful mathematical object (its distribution is the arcsine law for Brownian motion) but it cannot be used as a trading trigger.

Stopped processes

Given an adapted process

(X_n)

and a stopping time

\tau

, the stopped process is

X_n^\tau := X_{n \wedge \tau}.

After time

\tau

X_n^\tau

stays at its value at time

\tau

. Key fact: if $(X_n)$ is a martingale, so is $(X_n^\tau)$ (Doob's stopping theorem). The stopped process inherits the martingale property — this is the key mechanism that lets the optional stopping theorem work.

Discrete-time proof (one line):

X_{n+1}^\tau - X_n^\tau = (X_{n+1} - X_n)\cdot \mathbf{1}_{\tau > n}

, and

\mathbf{1}_{\tau > n}

\mathcal{F}_n

-measurable (since

\tau

is a stopping time). So conditionally on

\mathcal{F}_n

\mathbb{E}[X_{n+1}^\tau - X_n^\tau \mid \mathcal{F}_n] = \mathbf{1}_{\tau > n}\cdot \mathbb{E}[X_{n+1} - X_n \mid \mathcal{F}_n] = 0.

Done.

Why stopping times are the right abstraction

Four reasons stopping times earn their prominence in the stochastic-processes toolkit:

They formalise "no peeking." Any theorem about "whenever you stop, the process is fair" naturally requires the stop to be a stopping time.
Stopped processes preserve structure. Martingales, submartingales, local martingales — all remain in class under stopping. This is what makes localisation work: to deal with an unbounded or exploding process, stop it at a carefully chosen $\tau_n \uparrow \infty$ , prove things for the bounded stopped process, then pass to the limit.
First hitting times describe boundary events. Barrier options, lookback options, default times in credit modelling, triggers in pairs trading — all are first hitting times. Closed-form formulas for their distributions come from martingale methods applied at the stopping time.
Stopping times are what make stochastic integration well-defined in the Itô / semimartingale theory. Without them, the move from bounded integrands to the standard $L^2$ theory is impossible.

Common confusions and pitfalls

"Any random time I choose is a stopping time." No. Only times that are decidable from the past and present. The classic mistake: "let

\tau

be the time I would have bought at the minimum in hindsight." Not a stopping time. Not a legitimate trading rule. The optional stopping theorem does not apply.

"The $\sigma$ -algebra $\mathcal{F}_\tau$ is the natural filtration of $X_{n \wedge \tau}$ ." Nearly — but be careful:

\mathcal{F}_\tau

is the information available at the random time $\tau$ , which is a more subtle object than "the information available at some deterministic time." For stopping times,

\mathcal{F}_\tau

behaves well; for other random times it may not even be defined.

"Stopping times are closed under addition." Only in restricted senses.

\tau + c

for a deterministic constant

c

is a stopping time.

\sigma + \tau

for two stopping times can fail unless

\sigma

is decided first. Think carefully about what information is needed to know

\sigma + \tau \le n

"First hitting time of an open set is a stopping time in continuous time." Actually true, but it requires right-continuity of the filtration. In left-continuous or "bare" filtrations, hitting open sets can fail to be a stopping time. Always work with right-continuous filtrations for continuous-time hitting arguments.

"Stopping times are always finite." Not necessarily. A random walk's first hitting time of a high level may be

\infty

with positive probability; the optional stopping theorem requires additional integrability conditions in that case. The convention

\tau = \infty

must be explicitly handled.

"Optional stopping says $\mathbb{E}[M_\tau] = \mathbb{E}[M_0]$ always." Only under specific hypotheses (bounded

\tau

, bounded

M

, or

\mathbb{E}[\tau] < \infty

with bounded increments). The classic failure: for a symmetric random walk starting at

1

and stopping at

\tau = \inf\{n : S_n = 0\}

, we have

\mathbb{E}[S_\tau] = 0 \ne 1 = \mathbb{E}[S_0]

. Be careful.

Where this goes next

Optional Stopping Theorem: The payoff theorem for everything in this lesson.
Martingales (Discrete Time): The class of processes to which stopping times apply most powerfully.
Brownian Motion: Hitting times for Brownian motion have closed-form distributions (reflected-Brownian reflection principle, Lévy's arcsine law) that are core tools in barrier-option pricing.
American Options and Optimal Stopping: The decision of when to exercise an American option is the canonical optimal-stopping problem in finance. Solved by dynamic programming and by continuation-value algorithms (Longstaff-Schwartz).
Barrier Options: Knock-in and knock-out options pay off conditionally on a first hitting time, so their prices involve expectations of martingales stopped at hitting times.

References

Lawler, G. F. (2023). Stochastic Calculus: An Introduction with Applications. Ch. 1 §1.3 (Optional sampling theorem), Ch. 4 §4.1 (Martingales and local martingales).