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Unformatted text preview: J. Wissel Financial Engineering with Stochastic Calculus I Fall 2010 Assignment Sheet 3 1. Let X be a random variable on a probability space (Ω , F ,P ) with E [ X 2 ] < ∞ and G ⊂ F a sub σalgebra. a) Show that Y * := E [ X G ] satisfies E [( X Y * ) 2 ] ≤ E [( X Y ) 2 ] (1) for all Gmeasurable random variables Y with E [ Y 2 ] < ∞ . Moreover, show that if we have equality in (1) for some Y , then Y = Y * a.s. Hint. Consider E [( X Y ) 2 ( X Y * ) 2 G ]. Remark. This result says that the conditional expectation E [ X G ] minimizes the variance of the error Y X among all Gmeasurable estimates Y for X . b) Let Y be a Gmeasurable random variable with E [ Y 2 ] < ∞ . Show that Cov X E [ X G ] ,Y = 0. 2. a) Let M ( t ), t ≥ 0 be a martingale for a filtration ( F t ) t ≥ with E M ( t ) 2 < ∞ for all t ≥ 0. Show that E ( M ( t ) M (0) ) 2 = m 1 X j =0 E ( M ( t j +1 ) M ( t j ) ) 2 for any times 0 = t < t 1 < ... < t m = t ....
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This note was uploaded on 01/25/2011 for the course ORIE 5600 at Cornell.
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