stoppingtime

Stopping time is a concept in probability theory and stochastic processes describing a random time at which a decision to stop is made based on information up to that time. Formally, let (F_t) be a filtration, representing the information available by time t. A random variable tau taking values in [0, ∞] is a stopping time with respect to (F_t) if for every t ≥ 0 the event {tau ≤ t} belongs to F_t. This enforces a non-anticipation property: one can decide to stop at time t using only information available up to t.

In discrete time, tau is often integer-valued and satisfies {tau = n} ∈ F_n for each n. In continuous time, the condition is {tau ≤ t} ∈ F_t for every t ≥ 0. Stopping times can be finite or infinite, and may be bounded or unbounded.

Common examples include hitting or first-passage times. For a stochastic process X, the hitting time of a set A is tau_A = inf{t ≥ 0 : X_t ∈ A}. Classic instances are the first time a Brownian motion reaches a or the first time a simple random walk hits a specified level.

Key results and concepts related to stopping times include the optional stopping theorem, which provides conditions under which the expected value of a martingale at a stopping time equals its initial expectation. These conditions typically require certain integrability or boundedness assumptions on the stopping time and the process.

Applications of stopping times span various fields: pricing and exercise strategies for American options in finance, sequential analysis, and the broader area of optimal stopping, where one seeks to choose a stopping time to optimize an expected payoff.