Joint distribution of integrals of diffusion and driving noise

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, 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.

• How does Feynman-Kac give the distribution of $J$? In the form I am familiar with, it gives the expected value of J. – Nawaf Bou-Rabee Oct 28 '16 at 21:50
• For example, in Wikipedia's formulation of F-K (en.wikipedia.org/wiki/Feynman-Kac_formula), I would set $f=0$, $\psi=1$, and $V(x)=\mu x$, and thus potentially obtain the Laplace transform. – user78370 Oct 28 '16 at 21:55

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 $J$.