Is it possible for 2 Brownian Motions to have non-zero correlation without being jointly normal?
I'm a bit confused by the question.
I just assumed we always talk of multiple Brownian Motions as being jointly normally distributed with a certain covariance matrix etc.
I know in general two normal random variables have no reason a priori to be jointly normal. But does this also apply to Brownian motions which are a continuous stochastic process in which each B_i(t) is normally distributed with mean 0 and variance t for a given time t?
Consider the following stochastic differential equation (SDE):
$dB_t = \rho(B_t, W_t)dW_t + \sqrt{1 - \rho^2(B_t, W_t)}dZ_t$, where $\{W_t\}_t$ and $\{Z_t\}_t$ are two independant Brownian motions, and $\rho$ is a function taking values in $[-1, 1]$. Then the solution $\{B_t\}_t$ of this SDE is a Brownian motion, as $d\langle B, B\rangle_t = \rho^2(B_t, W_t)dt + (1 - \rho^2(B_t, W_t))dt = dt$ and the joint law of $\{(B_t, W_t)\}_t$ has no reasons to be a Gaussian distribution.
You can have a look at the third chapter of Damien Bosc, Three essays on modeling the dependence between financial assets, PhD thesis, 2012.