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

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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

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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 ) = μ ( t i + 4 t i ) + σ ( W ( t i + 4 t i )) - μt i - σW ( t i ) = μ 4 t i + σ 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 )) = μ 4 t i , V ar ( 4 G ( t i )) = σ 2 4 t i . Cov ( 4 G ( t i ) , 4 G ( t j )) = Cov ( σ 4 W ( t i ) , σ 4 W ( t j )) = σ 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 ) δ i,j = 4 t i δ i,j (2) where δ i,j is the Kronecker delta. (b) 0 0.5 1 1.5 2 04 02 0 02 04 Diffusion Simulated1.1 t, Time State 1 State 2 State 3 State 4 Figure 1: G(t) 2
2 Simple Poisson Process with Non Unit Am- plitude (a) Proof: 4 H ( t i ) = H ( t i + 4 t i ) - H ( t i ) = ν ( P ( t i + 4 t i )) - νP ( t i ) = ν 4 P ( t i ) (3) and E ( 4 P ( t i )) = V ar ( 4 P ( t i )) = λ 4 t i ; So, E ( 4 H ( t i )) = νλ 4 t i , V ar (

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

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