Results 1 to 2 of 2

Math Help - Geometric random walk with Normal steps

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    4

    Geometric random walk with Normal steps

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by matthew2040 View Post
    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.
    Y_k=\log(X_0)+(r_1+r_2+..r_k)-\log(X_0) =r_1+r_2+..r_k

    the sum of k iid RV \sim N(\mu.\sigma^2)

    CB
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Random Walk
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: October 18th 2010, 04:29 PM
  2. an almost random walk SP
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: July 15th 2010, 10:58 AM
  3. Random Walk in 1-D
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: May 3rd 2010, 09:14 AM
  4. random walk
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: November 16th 2009, 01:50 PM
  5. Random walk
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: February 19th 2009, 05:22 AM

Search Tags


/mathhelpforum @mathhelpforum