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TL;DR

A random variable \( \tau \) such that the event \( \{\tau \leq t\} \) depends only on information available at time t. Intuitively, you can decide whether to stop based only on what has happened so far, not the future.

By Valenke Exam Prep Team·Last updated 2026-06-03

Stopping Time

A random variable \( \tau \) such that the event \( \{\tau \leq t\} \) depends only on information available at time t. Intuitively, you can decide whether to stop based only on what has happened so far, not the future.

Why it matters for interviews

Stopping times formalize the notion of optimal decision-making in trading (when to exercise an option, when to close a position). The optional stopping theorem connects stopping times to martingale theory.

Definition and Mathematical Foundation

A random variable \( \tau \) such that the event \( \{\tau \leq t\} \) depends only on information available at time t. Intuitively, you can decide whether to stop based only on what has happened so far, not the future.

Application in Quantitative Finance

Stopping times formalize the notion of optimal decision-making in trading (when to exercise an option, when to close a position). The optional stopping theorem connects stopping times to martingale theory.

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Frequently Asked Questions

What is the secretary problem?▼
An optimal stopping problem: interview n candidates sequentially, deciding irrevocably to accept or reject each. The optimal strategy is to reject the first \( n/e \) candidates, then accept the next one better than all previous. Success probability converges to \( 1/e \approx 36.8\% \).
How do stopping times relate to American options?▼
An American option's value is the supremum over all stopping times of the expected discounted payoff: \( V = \sup_\tau E[e^{-r\tau} h(S_\tau)] \). Finding the optimal exercise time is an optimal stopping problem.
What conditions does the optional stopping theorem require?▼
The stopping time must be almost surely finite (or bounded), and the martingale must satisfy integrability conditions. Without these, the theorem can fail, leading to apparent paradoxes.