# affine function of random variable

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.

## migrated from quant.stackexchange.comAug 21 at 10:50

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• 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." – whuber Aug 21 at 14:42
• @whuber Thank you for your question - please see my edited question above. Hope the context is clearer now. – ilovevolatility Aug 21 at 15:57