I have to show that, if $W_t$ is a 1-d Brownian motion then $\biggl(W_t, \int_0^t W_s ds\biggr)$ has normal distribution. Hint: apply Ito formula to this bivariate process. Any idea or suggestion on how to solve it?
I tried to show that with characteristic function approach, since the marginal distributions have both normal distribution
If I want to show that the couple is bivariate gaussian I have to prove that , $\forall \lambda_1 , \lambda_2 \in \mathbb{R}$ $Z_t=\lambda_1 W_t + \lambda_2 \int_0^t W_s ds$ is normal, so $dZ_t=\lambda_1dW_t+\lambda_2W_tdt$ and if I compute $\phi_{Z_t}(\eta)$ and $d(\exp{i\eta Z_t})$ in the end I get $\phi_{Z_t}(\eta)=\int_0^t \mathbb{E}(\exp{(i\eta Z_s)} \cdot i\eta\lambda_2 W_s)ds-\int_0^t \mathbb{E}(\phi_{Z_s}) \cdot \eta^2 \cdot 1/2 \cdot ds$ so I don't know how to solve the first integral.