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ass4_W2012 - STAT/ACTSC 446/846 Assignment#4(due 1 The...

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Unformatted text preview: STAT/ACTSC 446/846 Assignment #4 (due March 20, 2012) 1. The Ornstein-Uhlenbeck process { X t } t ≥ is the unique solution to the equation dX t =- αX t dt + dB t , X = x. Show that X t = e − αt x + e − αt ∫ t e αs dB s and use this expression to calculate the mean and variance of X t . Hint: for the first part use the Ito’s lemma and for the second basic properties of the stochastic integral. 2. Let { B t } t ≥ be standard Brownian motion. Generate three trajectories of the process X n ( i n ) , i = 1 ,...,n , and show them on a graph, where { X n } is defined as follows: (a) X n ( i n ) = X n ( i − 1 n ) + 1 √ n Z i , Z i- i.i.d. binomial random variables taking values 1 and- 1 with probabilities p = 0 . 5 and q := 1- p = 0 . 5, respectively, X n (0) = 0, and n = 150. (b) X n ( i n ) = X ( i n ), where X ( t ) satisfies the following SDE equations (use n = 150): (i) Cox, Ingersoll, and Ross model: dX t = 2(0 . 3- X t ) dt + 0 . 1 √ X t dB t , X (0) = 0 . 4 ....
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ass4_W2012 - STAT/ACTSC 446/846 Assignment#4(due 1 The...

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