
covariance
Hi, I'm really lost..
Let W(t) be a standard Brownian Motion,
$\displaystyle X_t = t + 2^{t}W(4^t), t>=0 $
Find the covariance function of the process $\displaystyle X_t $
here is my working out, but im not sure if im doing it right, or what i should do next:
$\displaystyle cov (X_t,X_s) $
$\displaystyle = cov (t,X_s) + cov (2^{t}W(4^t), X_s) $
$\displaystyle = t cov(1,X_s) + 2^{t}cov(W(4^t), X_s) $
$\displaystyle = 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) $
$\displaystyle = 2^{t} cov(W(4^t),X_s) $
and now i don't know what to do, please help
here is the answer:
$\displaystyle R(t,s) = Cov(X_t,X_s) = 2^{ts} $