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Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach by Hong Xie, Chaoqun Ma, Guojun Gan

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CHAPTER 29

MARKOV PROCESSES

A Markov process is a stochastic process that has the Markov property. In other words, a stochastic process is called a Markov process if at every time t, the conditional probability law of the process given the past depends only on the present state. Intuitively, a Markov process is a process that does not have memory. In this chapter, we present the mathematical definition of Markov processes and relevant results.

29.1 Basic Concepts and Facts

Definition 29.1 (Conditional Independence). Let (Ω, , P) be a probability space. Let be sub-σ-algebras of . Then are said to be conditionally independent given if

equation

where Vi is an arbitrary positive random variable in mi, i = 1, 2,…, n. Here mi is the set of all i-measurable functions.

Definition 29.2 (Markov Process). Let ...

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