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Unformatted text preview: • Basic properties of conditional expectation. Let G be a σ-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 σ-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 rv’s B t i- B t i- 1 , i = 1 ,..,k (increments) are independent, (ii) B t- B s ∼ N ( μ ( t- s ) , σ 2 ( t- s )), where t > s , μ ∈ R , σ > 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....
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