used to denote the process X stopped at time τ. Then, X is said to locally satisfy some property P if there exists a sequence of stopping times τn, which increases to infinity and for which the processes satisfy property P. Common examples, with time index set I = [0, ∞), are as follows: Local martingale process. A process X is a local martingale if it is càdlàg and there exists a sequence of stopping times τn increasing to infinity, such that is a martingale for each n. Locally integrable process. A non-negative and increasing process X is locally integrable if there exists a sequence of stopping times τn increasing to infinity, such that for each n. 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 τ is predictable if it is equal to the limit of an increasing sequence of stopping times τn satisfying τn < τ whenever τ > 0. The sequence τn is said to announce τ, and predictable stopping times are sometimes known as announceable. Examples of predictable stopping times are hitting times of continuous and adapted processes. If τ 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 τn, where τn 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 τ is accessible if, P(τ = τn for some n) = 1, where τn are predictable times. A stopping time τ is totally inaccessible if it can never be announced by an increasing sequence of stopping times. Equivalently, P(τ = σ < ∞) = 0 for every predictable time σ. Examples of totally inaccessible stopping times include the jump times of Poisson processes. Every stopping time τ 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 τ = σ whenever σ < ∞, τ = υ whenever υ < ∞, and τ = ∞ whenever σ = υ = ∞. Note that in the statement