# covariance

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• November 9th 2009, 02:30 PM
lbmpa
covariance
Hi, I'm really lost..

Let W(t) be a standard Brownian Motion,
$X_t = t + 2^{-t}W(4^t), t>=0$

Find the covariance function of the process $X_t$

here is my working out, but im not sure if im doing it right, or what i should do next:
$cov (X_t,X_s)$
$= cov (t,X_s) + cov (2^{-t}W(4^t), X_s)$
$= t cov(1,X_s) + 2^{-t}cov(W(4^t), X_s)$
$= t cov (1,X_s) + ts cov(1,X_s) + 2^{-t} cov(W(4^t),X_s) + 2^{-t} scov(W(4^t),1)$
$= 2^{-t} cov(W(4^t),X_s)$

and now i don't know what to do, please help

here is the answer:
$R(t,s) = Cov(X_t,X_s) = 2^{-|t-s|}$