Consider a generic diffusion of the form
$$dX_t=f(t,X_t)dt+dB_t,$$
where $f$ is some *nice* function and $B_t$ is a standard Brownian motion.

The marginal distributions of the integrals
$$I:=\int_0^TB_t~dt\qquad J:=\int_0^TX_t~dt$$
can in principle be computed with fairly straightforward methods:
$I$ is Gaussian by a [classical Riemann sum argument][1],
and $J$ can (again, in principle) be computed by Feynman-Kac.

> **Question.** What about the joint distribution of $(I,J)$?

Are there general techniques that are well adapted to the solution of such problems? I'm also interested in any kind of nontrivial example where such computations have been made.


  [1]: https://math.stackexchange.com/questions/243925/integral-of-brownian-motion-is-gaussian