I wish to express as a Lebesgue integral the following expectation,
$E[\varphi(B_t)\varphi(B_s)]=\int ?$
for $0\leq s\leq t$, where $B_t$ is a Brownian motion with law $N(0,\sigma^2 t)$. Any ideas? I guess the point is to use independent increments since I would like to avoid the joint distribution.
Thank you very much! :)