StoppingTime

A stopping time is a random variable used in the context of stochastic processes, particularly in the study of probability theory and statistics. Stopping times are used to represent the time at which a certain event or condition is first satisfied in a stochastic process.

Formally, a stopping time τ with respect to a filtration 𝔽ₜ (a collection of increasing 𝜎-algebras representing the information available up to time t) is a random variable that takes on non-negative real values and satisfies the condition:

{τ ≤ t} ∈ 𝔽ₜ for all t ≥ 0

In simpler terms, this condition means that the event of the stopping time being less than or equal to a certain time t can be determined using the information available up to time t.

Stopping times are often used to model the occurrence of events in various applications such as finance, insurance, queuing theory, and many other fields. Some common examples of stopping times include the first time a stock price reaches a certain level, the time at which a queue exceeds a certain length, or the time until a gambler goes bankrupt.