If $X$ is a random variable, then $F(X) = a + bX$ where $a,b$ are constants is an affine function.
What is the technical/formal name of a function which is not affine, but for which the following always holds:
$$ dF(X) = g(\cdot) dX $$
i.e. the changes of the function $dF$ is always directly proportional to $dX$?
An example of such a function would be $$ F(X) = N(X) - N(-X) $$ with $N(\cdot)$ the normal CDF.
EDIT:
To give some more background:
I am looking at this problem in the context of quantitative finance / stochastic processes.
For affine functions, we have the following equality:
$$ E[F(X)] = F(E[X]) $$
Now I don't necessarily have an affine function, but I have a function $F(\alpha, X)$, where $\alpha$ is some parameter which I can always `tweak' such that the second derivatie of $F$ with respect to $X$ vanishes. Then according to the Ito-Doeblin formula I get
$$ dF(X) = \frac{\partial F}{\partial X} dX $$
which to me looks like making a non-affine function affine by varying a parameter (in my example above the "$\alpha$").
So I am interested in Jensen's (in)equality for these type of functions.
