Given two random variables $X$ and $Y$ and let $K$ be a constant value. Assume the expectation $\mathbb{E}[X(Y-K)^{+}]$ is given for all possible values of $K\geq 0$. Is there a way to derive the joint probability distribution of $X$ and $Y$ from this??
The expectation can be written as
$$\mathbb{E}[X(Y-K)^{+}]=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x(y-K)^{+}dF(x,y)$$ and when density exists $$=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x(y-K)^{+}f(x,y)dxdy$$
Both marginal distributions $F_{X}$ and $F_{Y}$ are known and densities exists as well. Is there any way I can derive the joint distribution if the expected value is given for all values of $K$?
I have been stuck on this for a while now, even rough approximations would be of much use to me.
Can someone please help me?