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Introductory Stochastic Analysis for Finance and Insurance by Society of Actuaries, X. Sheldon Lin

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CHAPTER 3

DISCRETE-TIME STOCHASTIC PROCESSES

3.1   STOCHASTIC PROCESSES AND INFORMATION STRUCTURES

Let images be a probability space. Recall that images is a collection of all possible events and represents all the information contained in the probability space. Imagine that a series of experiments is performed at times t = t0, t1, t2, ··· . For simplicity we assume in this chapter that ti = i, i.e., t = 0, 1, 2, ··· , unless otherwise specified. Let images be the collection of all possible events in images that may occur before or at time t. Thus images represents the information up to time t. Obviously,

(i)  images is an information structure coarser than images, i.e., images ⊆ , since it contains no more information than ;

(ii)  If

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