Homework2sol.pdf - IEOR-E4707 Spring 2018 Homework 2...

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IEOR-E4707, Spring 2018: Homework 2 Solutions February 18, 2018 Problem 1. Attempt the following problems a) Compute E [ W 4 t ]. b) Compute the mean and variance of R t 0 W s dW s . c) Some of the hyperbolic functions are sinh x = 1 2 ( e x - e - x ) , cosh x = 1 2 ( e x + e - x ) , tanh x = e x - e - x e x + e - x Let X satisfy the following equation dX t = 2 tanh X t dt + 2 dW t . Show that tanh X t is a martingale. Solution:
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b) E [ R t 0 W s dW s ] = 0 since E [ R t 0 W 2 s ds ] = E [ R t 0 sds ] < ; V ar ( R t 0 W s dW s ) = E [( R t 0 W s dW s ) 2 ] = E [ R t 0 W 2 s ds ] = 1 2 t 2 where the last equality follows from Ito’s isometry. c) First recall (tanh) 0 ( x ) = 1 - tanh 2 ( x ) and (tanh) 00 ( x ) = - 2 tanh x (1 - tanh 2 ( x )). Also, d h X i = 2 dt . Then from (Ito) d tanh X t =(1 - tanh 2 ( X t ))[2 tanh X t dt + 2 dW t ] + 1 2 [ - 2 tanh X t (1 - tanh 2 ( X t ))]2 dt = 2(1 - tanh 2 ( X t )) dW t integrating we get tanh X t = tanh X 0 + 2 Z t 0 (1 - tanh 2 ( X s )) dW s which is a martingale since tanh X t is uniformly bounded. Problem 2. Define X t = e 2 W t on [0 , T ]. Find the measure change d ˜ P d P that turns the process { X t } t [0 ,T ] into a martingale.

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