Question: 5. (10 points) Let Sn = EX. (n21) i=1 be a random walk on R where X1, X2, . .. is a sequence of i.i.d.

5. (10 points) Let Sn = EX. (n21) i=1 be a random walk on R where X1, X2, . .. is a sequence of i.i.d. normally distributed random variables with E[X,] = 0 and var(X, ) = o' e (0, co) (n = 1, 2,...). Define the sequence of random variables (Zn)n=1,2,... via Zn = exp(Sn - no?/2). (1)(a) Compute the expected value E[Z,] for all n 2 0. (b) Show that E[Zn+1 |S,] = Zn (n20). Carefully justify each step of your computations

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