Given a joint characteristic function, find $P(XThis question was asked in here before for a given MGF with discrete r.v.. A partial solution for the generalized version of the problem was given in this link by expiTTp1z0. But that is not nearly enough.

Question:
A joint characteristic function of $(X,Y)$ is given, find $P(X<Y)$.

 A: Taking the example of the joint MGF from this question,
$$M_{X,Y}(t_1,t_2) = \frac{1}{2}e^{t_1+t_2} + \frac{1}{4}e^{2t_1+t_2} + \frac{1}{12}e^{t_2} + \frac{1}{6}e^{4t_1+3t_2}$$
The joint characterisitic function of $X$ and $Y$ is,
$$\phi_{X,Y}(t_1,t_2) = M_{X,Y}(it_1,it_2) = \frac{1}{2}e^{i(t_1+t_2)} + \frac{1}{4}e^{i(2t_1+t_2)} + \frac{1}{12}e^{it_2} + \frac{1}{6}e^{i(4t_1+3t_2)}$$
Define $Z = Y-X$, then the characteristic function of $Z$ is,
$$\phi_{Z}(t) = \phi_{X,Y}(-t,t) = \frac{1}{2} + \frac{1}{4}e^{-it} + \frac{1}{12}e^{it} + \frac{1}{6}e^{-it}$$
By the inverse fourier transform of $\phi_{Z}(z)$, we obtain pdf $f_{Z}(z)$,
$$\begin{align}f_{Z}(z) &= \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-itz}\phi_{Z}(t)dt\\\\
&= \frac{1}{2\pi}\left(\int_{-\infty}^{\infty}e^{-itz}(\frac{1}{2}+ \frac{1}{4}e^{-it} + \frac{1}{12}e^{it} + \frac{1}{6}e^{-it})dt \right)\\\\
&= \frac{1}{2\pi}\int_{-\infty}^{\infty}\left(\frac{1}{2}e^{-itz} + \frac{1}{4}e^{-it(z+1)}+\frac{1}{12}e^{-it(z-1)}+\frac{1}{6}e^{-it(z+1)}\right)dt\\\\
&=\frac{1}{2}\delta(z) + \frac{1}{4}\delta(z+1)+\frac{1}{12}\delta(z-1)+\frac{1}{6}\delta(z+1)\end{align}$$
where $\delta(z-c) = 1 \text{ when } z-c = 0$.
From here, one can easily find $F_{Z}(z)$.
Overall the steps are,


*

*Compute joint characteristic function $\phi_{X,Y}(t_1, t_2)$.

*Define $Z = Y - X$ and compute characteristic function of $Z$, $\phi_{Z}(t) = \phi_{X,Y}(-t,t)$.

*Compute the inverse fourier transform of $\phi_{Z}(t)$ to obtain $f_{Z}(z)$.

*Using $f_{Z}(z)$ compute the cdf $F_{Z}(z)$.


Thanks @drhab for this wiki link.
Helpful link: computing last integrals leading to dirac delta functions, link.
