Let X1, X2,....be a geometric random walk with normal steps. More formally, we assumet that, for any positive integer k,
Xk = X0exp(r1+.....rk)
where X0 is a fixed positive constant and r1, r2, .....are i.i.d N(μ,σ^2)
1. Determine the expectation and the variance of the random variable Yk = log(Xk/X0) in terms of k, μ and σ^2.
2. Let μk = kμ + log(X0) and σ^2 k = kσ^2. Then, show that Xk is Lognormal (μk, σ^2 k)
3. Determine the c.f.d and the p.d.f. of Xk
4. Show that the expectation of Xk is exp(μk + (σ^2 k)/2 ) = exp(kμ + log(X0) + k(σ^2/2)
Hint: Start with the definition of the expectation and use the change of variable y=log(x)-μk. You will also need the expression of the p.d.f. of the normal distribution with arbitrary parameters.
5. Show that E(X^2 k ) = {E(Xk)}^2 exp(kσ^2), and deduce the variance of Xk
Could you please walk me through the these questions.