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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.

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migrated from quant.stackexchange.com Aug 21 at 10:50

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  • 2
    $\begingroup$ Could you explain why you state that $X$ is a "random variable" rather than just a number? This question seems to be one about real-valued functions of real variables only, unless perhaps "$dX$" and "$dF$" have some special meaning for random variables--but what would that be? And if it's about real-valued functions, the answer is well-known: such functions are called "differentiable." $\endgroup$ – whuber Aug 21 at 14:42
  • $\begingroup$ @whuber Thank you for your question - please see my edited question above. Hope the context is clearer now. $\endgroup$ – ilovevolatility Aug 21 at 15:57

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