Types of Stopping Times
Stopping times, with time index set I = [0,∞), are often divided into one of several types depending on whether it is possible to predict when they are about to occur.
A stopping time is predictable if it is equal to the limit of an increasing sequence of stopping times n satisfying n < whenever > 0. The sequence n is said to announce, and predictable stopping times are sometimes known as announceable. Examples of predictable stopping times are hitting times of continuous and adapted processes. If is the first time at which a continuous and real valued process X is equal to some value a, then it is announced by the sequence n, where n is the first time at which X is within a distance of 1/n of a.
Accessible stopping times are those that can be covered by a sequence of predictable times. That is, stopping time is accessible if, P(=n for some n) = 1, where n are predictable times.
A stopping time is totally inaccessible if it can never be announced by an increasing sequence of stopping times. Equivalently, P( = σ < ∞) = 0 for every predictable time σ. Examples of totally inaccessible stopping times include the jump times of Poisson processes.
Every stopping time can be uniquely decomposed into an accessible and totally inaccessible time. That is, there exists a unique accessible stopping time σ and totally inaccessible time υ such that = σ whenever σ < ∞, = υ whenever υ < ∞, and = ∞ whenever σ = υ = ∞. Note that in the statement of this decomposition result, stopping times do not have to be almost surely finite, and can equal ∞.
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