This preview shows page 1. Sign up to view the full content.
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 i B t i 1 , i = 1 ,..,k (increments) are independent, (ii) B t B s N ( ( t s ) , 2 ( t s )), where t > s , R , > 0. Definition . A process M = { M t , t } adapted to {F t , t } is called a martingale iff E ( M t F s ) = M s for all s < t....
View
Full
Document
 Spring '09
 Adam

Click to edit the document details