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Unformatted text preview: STAT/ACTSC 446/846 Assignment #4 (due November 23, 2009) 1. Let { W t } be defined by the SDE dW t = μdt + σdB t , where B t is a standard Brownian motion. Use Ito’s formula to write the following processes Y t in the form of a stochastic integral ( dY t = u ( Y t ,t ) dt + v ( Y t ,t ) dB t or an equivalent integral form): (a) Y t = W 3 t , t ≥ (b) Y t = 10 + t 2 + e 4 W t , t ≥ (c) Y t = exp ( W 2 t σt ), t ≥ 0. 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 = 3(0 . 2 X t ) dt + 0 . 06 p X t dB t , X (0) = 0...
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This note was uploaded on 01/11/2010 for the course ACTSC 446 taught by Professor Adam during the Fall '09 term at Waterloo.
 Fall '09
 Adam

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