I How to find the inverse of an integral transform?

[hyurnat4]

I'm trying to find the distribution of a random variable $T$ supported on $[t_1, t_2]$ subject to $\mathbb{E}[V(t', T)] = K, \forall t' \in [t_1, t_2]$. In integral form, this is : $$ \int_{t_1}^{t_2} V(t', t).f(t) \, dt = K,\forall t' \in [t_1, t_2], $$ which is just an exotic integral transform.

So if I can find the inverse transform, I'm done. But the function $V$ is ...not nice. It's nowhere differentiable for $t < t'$, there's a jump at $t = t'$ and it's constant from then on.

So do you guys know of any methods to find the inverse to an integral transform with a nasty kernel like $V$? Or can you see another way to solve these equations? I'd even be happy with numerical techniques.

Thanks in advance.

 

[soarce]

$V(t^\prime,t)$ is similar to Brownian motion for $t<t^\prime$?