homework - CNETzhengh HW1 Zheng Hong October 5, 2009...

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Unformatted text preview: CNETzhengh HW1 Zheng Hong October 5, 2009 Contents 1 Gaussian Process from Zero Mean, Unit Variance Wiener Pro- cess 2 2 Simple Poisson Process with Non Unit Amplitude 3 3 Wiener and Poisson Tables Partial Justification 4 4 Integral of Wiener Differential Squared 6 5 Integral of Product of Time and Wiener Differentials 8 1 1 Gaussian Process from Zero Mean, Unit Vari- ance Wiener Process (a) Proof: 4 G ( t i ) = G ( t i + 4 t i )- G ( t i ) = ( t i + 4 t i ) + ( W ( t i + 4 t i ))- t i- W ( t i ) = 4 t i + 4 W ( t i ) (1) and E ( 4 W ( t i )) = 0, V ar ( 4 W ( t i )) = 4 t i ; So, E ( 4 G ( t i )) = 4 t i , V ar ( 4 G ( t i )) = 2 4 t i . Cov ( 4 G ( t i ) , 4 G ( t j )) = Cov ( 4 W ( t i ) , 4 W ( t j )) = 2 Cov ( 4 W ( t i ) , 4 W ( t j )) = E ( 4 W ( t i ) * 4 W ( t j )) = E (( 4 W ( t i )) 2 ) i,j = 4 t i i,j (2) where i,j is the Kronecker delta....
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This note was uploaded on 04/19/2010 for the course FIN 390 taught by Professor Hansen during the Fall '09 term at University of Illinois, Urbana Champaign.

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homework - CNETzhengh HW1 Zheng Hong October 5, 2009...

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