This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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...
View
Full
Document
This note was uploaded on 01/11/2010 for the course ACTSC 446 taught by Professor Adam during the Fall '09 term at Waterloo.
 Fall '09
 Adam

Click to edit the document details