Suppose that $W_{1,t}$ is a Brownian Motion and $W_{2,t}$ is another Brownian Motion. The two Brownian Motions have correlation equal to $\rho$.
Let C = $W_{1,t} \cdot W_{2,t}$
What is the distribution of $C$?
I did make this up - so no guarantees there's a solution...but I feel like there should be!
I was able to get $E(C) = \rho t$ and I know $dW_{1,t} \cdot dW_{2,t} = \rho dt$
Any advice? Is $C$ normally distributed? Can the variance be computed too?