Continuous_Time_-_Conditional_Expectation

Continuous_Time_-_Conditional_Expectation - 1 Temple...

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  1 Temple University Continuous Time Finance Conditional Expectation
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  2 Conditional Expectation 2.1 Information and σ-algebras
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  3 Borel σ-algebra of R The Borel algebra on the reals is the smallest σ-algebra on R which contains all the intervals. c n n n b a b a b a b a )) , ( ) , (( ] , [ ] , [ ) , ( 1 1 1 -∞ = - + = = } ) ( : { ) ( 1 B w X B X = - ϖ B
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  4 σ-algebra generated by X F F B} Ω:X(ω) B} {X B X:Ω P F algebra - in the is of subset Borel every for s.t. : function e) (measurabl valued - real a is variable random A ) , , ( space y Probabilit σ = )} ( : ) ( 1 - {X (X) ) Borel( B over ranges B where B} {X subsets of collection the (X) by Denote = Borel B B σ σ
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  5 X is F -measurable ) Borel( B over ranges B where B} {X subsets of collection the (X) by Denote σ measurable is X say We If F- F σ(X) The “information” F is sufficient to determine the value of X
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  6 Filtration (History) Filtration a ) ( call We t algebra - nonempty 0 + 2200 t t t s t F t if s F F F T σ Filtration represents the information available at each time t , and is more and more precise - the set of measurable events is staying the same or increasing - as information from the present becomes available. Nested σ-algebras
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  7 Adapted Stochastic Process 0 0 t ) filtration the to adapted is X ... say we 0, t measurable - is If variables random ) ( 0 t , of filtration a space sample nonempty t t t t t t (F F X X F Filtration reflects the information
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