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


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 ...

Get Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach now with the O’Reilly learning platform.

O’Reilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers.