Stopping times are important tools that allow us to analyze stochastic processes by viewing them at random times. In this chapter, we present a definition and relevant results of stopping times.

23.1 Basic Concepts and Facts

Definition 23.1 (Discrete Stopping Time). Let (Ω, , P) be a probability space and {n, n = 0, 1, 2, …} an increasing sequence of sub-σ-fields of . A stopping time for {n : n ≥ 0} is a function


such that {Tn} n for each n ≥ 0.

A stopping time for a sequence of random variables, {Xn}n≥0, is a stopping time relative to the σ-fields n = σ(X0, X1, …, Xn).

In the above definition, the index starts from 0. The definition can be modified in the obvious ...

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