I am revisiting my covariance rules so that I can have a better understanding of how does covariance of brownian motion works. Basically what I did was.
$Cov(X-Y,X-Y) = Cov(X,X) +2Cov(X,-Y)+Cov(-Y,-Y)=Cov(X,X)-2Cov(X,Y) +Cov(-Y,-Y)=Var(X) +Var(Y)-2Cov(X,Y)$.
However this doesn't match, $Cov(X-Y,X-Y) = Var(X-Y) = Var(X)+Var(Y)$. Can Some one explain what I did wrong. On another note: I'm finding stochastic processes really difficult compared to like say inference, modelling. Is this normal? The course I'm doing is probably the hardest course I've done at university(in Aust) to date.
$Var(X-Y)=Var(X)+Var(Y)-2Cov(X,Y)$. Only when X and Y are are uncorrelated when $Cov(X,Y)=0$ does the sum of the variances equal the variance of the difference.