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](https://math.stackexchange.com/questions/243925/integral-of-brownian-motion-is-gaussian),
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.



Joint distribution of integrals of diffusion and driving noise
Consider the SDE:
$$
\begin{cases}
d X\_t = f(t,X\_t) dt + d B\_t \\
d Y\_t = d B\_t
\end{cases}
$$
The infinitesimal generator of the process $(X\_t,Y\_t)$ is given by:
$$
L\_t g(x,y) = f(t,x) \partial\_x g(x,y) + \frac{1}{2} \partial\_{xx} g(x,y) + \partial\_{xy} g(x,y) + \frac{1}{2} \partial\_{yy} g(x,y) 
$$ Let $t\_1$ and $t\_2$ be real constants. At least formally, the solution to the PDE: 
$$
\partial\_t u\_t(x,y) = L\_t u\_t(x,y)+ (t\_1 x + t\_2 y) u\_t(x,y)\;, \quad \left. u \right|\_{t=0} = 1
$$ 
admits the stochastic representation
$$
u\_t(x,y)= \mathbb{E}\_x \left\{ \exp( t\_1 I + t\_2 J ) \right\}
$$ which we recognize as the *joint moment generating function* of $I$ and $J$. 


