Formulae - Basic properties of conditional expectation. Let...

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Unformatted text preview: Basic properties of conditional expectation. Let G be a -field and X and Y be integrable random variables. Then, a) E ( aX 1 + bX 2 |G ) = aE ( X 1 |G ) + bE ( X 2 |G ) b) E ( E ( X |G )) = EX c) E ( X |G ) = X if X is G measurable (depends on information contained in G ) d) E ( XY |G ) = Y E ( X |G ) if Y is G measurable e) E ( X |G ) = EX if X is independent of G f) Suppose that H G (i.e. H is a sub -field of G ). Then E [ E ( Y |H ) |G ] = E [ E ( Y |G ) |H ] = E ( Y |H ) Definition. A Brownian motion is a stochastic process { B t , t } such that B = 0 and (i) for any 0 t < t 1 < < t k , the rvs B t k- B t k- 1 (increments) are independent, (ii) B t- B s N ( ( t- s ) , 2 ( t- s )), where t > s , R , > 0. A filtration {F t , t } is generated by a process { X t , t } if F t is the smallest - field with respect to which all X s , s t , are measurable ( F t represents information generated by the process up to time t...
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This note was uploaded on 01/11/2010 for the course ACTSC 446 taught by Professor Adam during the Fall '09 term at Waterloo.

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Formulae - Basic properties of conditional expectation. Let...

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