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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 21

STOCHASTIC PROCESSES

A stochastic process is a collection of random variables. Stochastic processes play an important role in mathematical finance as they are used to model the stock prices. In this chapter, we shall present some general concepts and results about stochastic processes.

21.1 Basic Concepts and Facts

Definition 21.1 (Stochastic Process). Let I be a set and (E, ) a measurable space. A stochastic process indexed by I is a family of measurable functions Xt, t I, from a probability space (Ω, , P) into (E, ). The measurable space (E, ) is referred to as the state space.

For each ω Ω, the mapping tXt(ω) is called a trajectory or a sample path of X.

Definition 21.2 (Probability Law of Stochastic Process). Let {Xt : t I} be a stochastic process on some probability space ...

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