Suppose that X1,X2, . . . is a lognormal geometric random walk with parameters (, 2). More

Question:

Suppose that X1,X2, . . . is a lognormal geometric random walk with parameters (μ, σ2). More specifically, suppose that Xk = X0exp(r1 + ∙ ∙ ∙ +rk), where X0 is a fixed constant and r1, r2, . . . are i.i.d. N(μ, σ2).
(a) Find P(X2 > 1.3X0).
(b) Use (A.4) to find the density of X1.
(c) Find a formula for the 0.9 quantile of Xk for all k.
(d) What is the expected value of X2k for any k? (Find a formula giving the expected value as a function of k.)
(e) Find the variance of Xk for any k.