ma503b-10hm2

# ma503b-10hm2 - MATH503b (SPRING 2010) HOMEWORK #2 Due...

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MATH503b (SPRING 2010) HOMEWORK #2 Due Wednesday, February 17 1) Let H be a complex inner product space, whose inner product is deﬁned by ( x,y ) = x y , for all x,y H . Let k x k := ( x,x ) 1 / 2 . Prove the Cauchy-Schwartz inequality : | ( x,y ) | ≤ k x kk y k , x,y H . Using the Cauchy-Schwartz inequality to verify the triangle inequality : k x + y k ≤ k x k + k y k , x,y H . (Hint: To prove the Cauchy-Schwartz inequality, you might want to ﬁrst choose a complex number α , with | α | = 1, and α ( x,y ) = | ( x,y ) | , and then try to modify the argument in class.) 2) Assume T 1 > T 2 , and let M be a martingale such that E | M T | 2 < for all T > 0. Then by Martingale Representation Theorem, we can ﬁnd two processes H 1 L 2 T 1 ( B ) and H 2 L 2 T 2 ( B ), respectively, such that it holds almost surely, M t = M 0 + Z t 0 H 1 s dB s , t [0 ,T 1 ]; and M t = M 0 + Z t 0 H 2 s dB s , t [0 ,T 2 ] . Argue that the representation is “consistent” in the sense that H 1 t = H 2 t , almost surely, for all t [0 ,T 2 ].
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## This note was uploaded on 10/08/2010 for the course MA 503 taught by Professor Majin during the Fall '09 term at USC.

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