If we have a joint Laplace-Stieltjes transform (or MGF or characteristic function) of some independent nonnegative continuous random variables $X = (X_1, X_2, ..., X_n)$ defined on $[0,\infty)$, the LST would be $L(t)=\mathbb{E}\left[e^{-t\cdot X}\right]=L_{X_1}(t_1)\cdots L_{X_n}(t_n)$. Under certain conditions, we can do inverse Laplace transforms to find marginal PDFs, which we can multiply to get the joint PDF.
If the $X_j$'s are dependent random variables, we cannot carry out most of these steps. Is there some way to find the joint PDF from a Laplace-Stieltjes transform (or MGF or CF)?
I'm not sure if perhaps it can be done analytically if the LST is simple enough or if numerical inversions are required, or if some of these are approaches are viable under certain assumptions.
Thank you!