I have troubles concerning a correlated bivariate GBM with identical drift and diffusion rates.
Let $dX^i_t = \mu X^i_t dt + \sigma X^i_tdW^i_t$
and $E[dW_t ^idW^j_t] = \rho_{i,j}dt$
If $X_0^i = 1$, I already learned that for $s<t$
$Cov(X^i_s, X^i_t) = exp((\mu+0.5\sigma^2)\cdot(s+t)) \cdot (exp(\sigma^2 s)-1)$
and
$Cov(X^i_t, X^j_t) = exp(2\mu t)\cdot(exp(\sigma^2\rho t)-1)$
The former may be referred to as time series covariance and the latter as cross sectional covariance.
However for $s<t$, I'm stuck on the quantity:
$Cov(X^i_s, X^j_t) = ???$
I appreciate your comments!
I think I got it:
Since $Cov(X^i_s, X^j_t) = Cov[X^i_s, X^j_s\cdot exp((\mu-\sigma^2/2)(t-s)+\sigma(W(t)-W(s)))] $
it follows that
$Cov(X^i_s, X^j_t) = exp((\mu-\sigma^2/2)(t-s))\cdot Cov[X^i_s, X^j_s\cdot exp(\sigma W(t)-\sigma W(s))]$
and this yields
$Cov(X^i_s, X^j_t) = exp((\mu-\sigma^2/2)(t-s))\cdot Cov[X^i_s, X^j_s]$