Consider the following the integral equation: $$ f(s) = \int_a^b K(s,t) g(s,t) dt, $$ where $f$ and $K$ are known functions, and $g$ is the unknown function we want to solve for.
For example, $K(s,t)$ is the conditional density of a random variable $T$ given $S$. So $K \geq 0$ and $\int K(s,t)dt = 1$. Then the question becomes: if we know the conditional expectation $$ \mathbb{E}[g(S,T) \mid S] = f(S), $$ then can we back out the original function $g$?
Is there any way to solve this kind of problem? All the books I find talking about integral equations do not allow $g$ to be a function of $s$. I'm really curious about this more general situation where $g$ is a function of both $t$ and $s$ but couldn't find any reference to go to.
Thanks in advance for any help and consideration!