If we have a random variable $X(t)$, the evolution equation for its probability density function (PDF) $f(x, t)$ is given by the Kramers–Moyal expansion:
$$ \frac{\partial f(x,t)}{\partial t} = \sum_{n = 1}^\infty (-1)^n \frac{\partial^n}{\partial x^n} \bigl[ D^{(n)}(x, t) f(x,t) \bigr] $$
where the Kramers--Moyal coefficients are
$$ D^{(n)}(x,t) = \frac{1}{n!} \lim_{\Delta t \to 0} \langle [X(t+\Delta t) - X(t)]^n \mid X(t) = x \rangle $$
Question: What would be the respective evolution equation for the cumulative distribution function (CDF) $F(x, t)$?
$$ \frac{\partial F(x,t)}{\partial t} = {}? $$
It would be also nice if you consider the multivariate case.
I have an attempt at this. Below I assume that the functions are such that all the operations are correct. Starting from
$$ F(x, t) = \int_{-\infty}^x f(z,t) \, \mathrm dz $$
We can apply $\int_{-\infty}^z$ to both sides of the Karamers–Moyal expansion for $f(z,t)$:
$$ \frac{\partial F(x, t)}{\partial t} = \sum_{n=1}^{\infty} (-1)^n \int_{-\infty}^{x} \frac{\partial^n}{\partial z^n} \bigl[ D^{(n)}(z,t) f(z,t) \bigr] \, \mathrm dz $$
With the fundamental theorem of calculus and assuming that everything vanishes at minus infinity, finally we get:
$$ \frac{\partial F(x, t)}{\partial t} = \sum_{n=1}^{\infty} (-1)^n \frac{\partial^{n-1}}{\partial x^{n-1}} \bigl[ D^{(n)}(x,t) f(x,t) \bigr] \, \mathrm dx. $$