Chap2 - Chapter 2. Information and Conditioning Definition...

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Unformatted text preview: Chapter 2. Information and Conditioning Definition 2.1.1. Let be a nonempty set. Let T be a fixed positive number, and assume that for each t [0 , T ] there is a -field F ( t ). Assume further that if s t , then every set in F ( s ) is also in F ( t ). Then we call the collection of -fields F ( t ) , t T , a filtration. Definition 2.1.3. Let X be a random variable defined on a nonempty sample space . The - field generated by X , denoted ( X ), is the collection of all subsets of of the form { X B } , where B ranges over the Borel subsets of R . Definition 2.1.5. Let X be a random variable defined on a nonempty sample space . Let G be a -field of subsets of . If every set in ( X ) is also in G , we say that X is G-measurable. 1 Definition 2.1.6. Let be a nonempty sam- ple space equipped with a filtration F ( t ) , t T . Let X ( t ) be a collection of random variables indexed by t [0 , T ]. We say this collection of random variables is an adapted stochastic process if, for each t , the random variable X ( t ) is F ( t )-measurable....
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Chap2 - Chapter 2. Information and Conditioning Definition...

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