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fortissimo

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## Homework Statement

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?

## Homework Equations

Is this possibly in the right direction?

## 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.