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# Tut1 - ACTSC/STAT 446/846 Tutorial#1 Let W = cfw_Wt t 0 be...

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ACTSC/STAT 446/846 Tutorial #1 January 21, 2009 Let W = { W t , t 0 } be a standard Brownian motion. 1. Let Z 1 , Z 2 , . . . be positive iid random variables with mean 1 . Define X 0 = x and X n = Q n j =1 Z j for all n 1 . Show that X = { X n , n 0 } is a martingale with respect to the filtration generated by the Z n ’s and find its mean. 2. The price of a stock at time n is modelled by S n = S 0 e ε 1 + ··· + ε n , where S 0 is a positive constant and where the ε k ’s are i.i.d. random variables with common distribution ( a with probability q ; b with probability 1 - q . The continuously compounded risk-free rate is given by r . Find the value of q for which the discounted price D n = e - rn S n is a martingale with respect to F = {F n } n 0 , the filtration generated by the ε k ’s. If you need any restrictions on a, b and r , state and justify them. 3. Let B = { B t , t 0 } be the Brownian motion with drift μ and diffusion coefficient σ obtained as a linear transformation of W . (a) Prove that C ov ( W s , W t ) = min { s, t } ; (b) Compute the covariance of W s and B t ; (c) Compute E e B t . 4. Let X = ( X t ) 0 t 1 be the so-called Brownian bridge, that is, for 0 t 1 , define X t = W t - tW 1 . Compute the mean, variance and covariance functions of X . 5. Assume that S =

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