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Auto-covariance of a Wiener process of a function

  1. Sep 18, 2013 #1
    1. The problem statement, all variables and given/known data

    Let {W(t); t >= 0} be a Wiener process. Determine the auto-covariance function for the process {X(t); t >= 0} defined by X(t) = e^(-ct) * W(e^(2ct)) for all t >= 0, where c > 0 is a constant.

    Is {X(t); t >= 0} stationary in the wide sense?

    2. Relevant equations

    Is this possibly in the right direction?

    3. The attempt at a solution

    C_X (t,τ) = Cov(X(t), X(t+τ)) = Cov(e^(-ct) * W(e^(2ct)),e^(-c(t+τ)) * W(e^(2c(t+τ)))) = e^(-c(2t+τ))*Cov(W(e^(2ct)), W(e^(2c(t+τ)))) = α*e^(-c(2t+τ))*min{e^(2ct),e^(2c(t+τ))}.

    The process is not stationary in the weak sense since the auto-correlation function (equal to the auto-covariance here?) varies with t.
     
  2. jcsd
  3. Sep 19, 2013 #2
    I have solved the problem, no need to reply to this.
     
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