I have been grappling with a mathematical problem that I haven't been able to solve and I'm hoping someone here can provide some insight. I'm trying to find two functions, f and g, that satisfy the following equations:
$$f(z)= z + \frac{\mathbb{P}[X \leq g(z)]}{\frac{d}{dz}\mathbb{P}[X \leq g(z)]}$$
$$g(z)= z + \frac{\mathbb{P}[Y \leq f(z)]}{\frac{d}{dz}\mathbb{P}[Y \leq f(z)]}$$
where X and Y are random variables with density.
Even a proof of the existence of such functions would be a great help. I suspect that a fixed point theorem might be applicable here, but I'm not sure how to apply it...
Any guidance or references to relevant literature would be greatly appreciated!
Thank you!