Hong,Zheng HW3 - CNETzhengh HW3 Zheng Hong October 19, 2009...

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Unformatted text preview: CNETzhengh HW3 Zheng Hong October 19, 2009 Contents 1 Finance Oriented Martingales 2 2 Exponential Martingale 2 3 Trigonometric Ito Integral 3 4 Solve diffusion SDE 3 5 Square Root Noise Problem 4 1 1 Finance Oriented Martingales (a) Proof: M 1 ( t ) = LnX ( t )- E [ Ln ( X ( t ))], dX ( t ) = X ( t )( ( t ) dt + ( t ) dW ( t )); Y ( t ) = LnX ( t ); dY ( t ) = ( ( t )- 1 2 2 ( t )) dt + ( t ) dW ( t ); Y ( t ) = LnX ( t ) + R t t ( ( s )- 1 2 2 ( s )) ds + R t t ( s ) dW ( s ); E [ dY ( t )] = ( ( t )- 1 2 2 ( t )) dt ; M 1 ( t )- M 1 ( s ) = [ Y ( t )- Y ( s )]- E [ Y ( t )- Y ( s )] = Z t s ( ( x )- 1 2 2 ( x )) dx + Z t s ( x ) dW ( x )- E [ Z t s ( ( x )- 1 2 2 ( x )) dx + Z t s ( x ) dW ( x )] = Z t s ( x ) dW ( x )- E [ Z t s ( x ) dW ( x )] (1) So, M 1 ( t )- M 1 ( s ) and M 1 ( s ) are independent. E [ M 1 ( t ) | M 1 ( s )] = E [ M 1 ( t )- M 1 ( s ) | M 1 ( s )] + M 1 ( s ) = M 1 ( s ); That means, M 1 ( t ) is a martingale.) is a martingale....
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Hong,Zheng HW3 - CNETzhengh HW3 Zheng Hong October 19, 2009...

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