covariation of martingales

Question

For a bivariate brownian motion $(B_t,W_t)_t$ we have that the covariation is given by $\langle W,B\rangle _t = t\rho$ where $\rho$ is the (constant) correlation between $W$ and $B$. Does this hold particularly because for fixed $t$, $(B_t,W_t)$ are multivariate normal? If not is there a counter example when $B_t$ and $W_t$ are brownian motions which are not mutually normal?

Answer 1

an extension of Lévy's theorem on the characterization of Brownian motion in the multivariate case is given in George Lowther 's blog here theorem 2.

It asserts that if your process is a continuous local martingale with covariation of the type you state in your post then it has to be a multidimensional Brownian motion.

Now here is a "kind" of counterexample where only the marginals are Brownian motions. You specify the joint law by any non gaussian copula (depending on $t$) and the resulting process is not a bi-dimensional Brownian motion (and hence can't be a continuous local martingale otherwise it would contradict Lévy's characterization !!!)

Best regards