# CHAPTER 23

# STOPPING TIMES

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 {*T* ≤ *n*} _{n} for each *n* ≥ 0.

A stopping time for a sequence of random variables, {*X*_{n}}_{n≥0}, is a stopping time relative to the σ-fields _{n} = σ(*X*_{0}, *X*_{1}, …, *X*_{n}).

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

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