610hw3 - FE 610 Homework 3 Lena Yemelyanov Due Wednesday...

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Unformatted text preview: FE 610 Homework 3 Lena Yemelyanov Due: Wednesday, November 8, 2009 The Full Assignment is from PDF available online Question 1 : Problem statement : Differentiate the following functions with respect to the Wiener process W t , and if applicable, with respect to t. (a) f W t W t 2 f W t W t Solution for ( a ): dF W t , t F s dW t F t dt 1 2 F ss dt F s- first derivative of the innovation term F ss- second derivative of the innovation term F t- first derivative of teh drift term f W t W t 2 F s dW t 2 W t dW t Since our f W t does not have a drift term F t dt 1 2 F ss dt 1 2 ∗ 2 ∗ dt dt f ′ W t dt 2 W t dW t The second problem is solved in a similar manner as the problem above, since the only difference is the exponent. f W t W t W t 1/2 F s dW t 1 2 W t − 1/2 dW t dW t 2 W t F t dt 1 2 F ss dt 1 2 ∗ 1 2 ∗ − 1 2 ∗ W t − 3/2 dt − 1 8 W t − 3/2 dt f ′ W t − 1 8 W t − 3/2 dt 1 2 W t − 1/2 dW t Problem statement ( b ): f W t e W t 2 Solution ( b ): 1 dF W t , t F s dW t F t dt 1 2 F ss dt F s dW t 2 W t e W t 2 dW t F t dt 1 2 F ss t 2 dt 1 2 2 e W t 2 4 W t 2 e W t 2 dt e W t 2 dt 2 W t 2 e W t 2 dt dF W t , t e W t 2 2 W t 2 e W t 2 dt 2 W t e W t 2 dW t Problem statement ( c ): f W t , t e W t − 1/2 2 t f W t , t e W t Solution ( c ): f W t , t e W t − 1/2 2 t dF W t , t F s dW t F t dt 1 2 F ss dt F s dW t e W t − 1/2 2 t dW t F t dt − 1 2 2 e W t − 1/2 2 t dt 1 2 F ss dt 1 2 2 e W t − 1/2 2 t dt dF W t , t e W...
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This note was uploaded on 02/27/2010 for the course FE 610 taught by Professor Prasad during the Spring '10 term at Stevens.

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610hw3 - FE 610 Homework 3 Lena Yemelyanov Due Wednesday...

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