You'd probably get more responses if you posted in LaTex and stated the question concisely. As I interpret the question, a change of variables would reduce it to the following:
Given the following:
The function [itex]S(t)[/itex] satisfying [itex]S(0) = 1 ; S(1)= 0[/itex] and [itex]\frac{dS}{dt} < 0[/itex] for [itex]0 < t < 1[/itex]
Find the following:
Constants [itex]X_{min} < X_{max}[/itex]
A random variable [itex]X[/itex] that has support [itex][X_{min}, X_{max}][/itex].
and a function [itex]F(S,X)[/itex]
that will satisfy these conditions:
[itex]F(S(0),x) = 1[/itex] for all [itex]X_{min} \leq x \leq X_{max}[/itex]
[itex]F(S(1),x) = 0[/itex] for all [itex]X_{min} \leq x \leq X_{max}[/itex]
[itex]\frac{\partial F(S,x)}{\partial x } < 0[/itex] for [itex]0 < t < 1[/itex]
For each [itex]0 < t < 1[/itex], [itex]\int_{X_{min}}^{X_{max}} F(S(t),X) dX = S(t)[/itex]
You didn't give a specific [itex]S[/itex] so I assume an answer must be expressed in terms of [itex]S[/itex]. You used the constant [itex]t_0[/itex] in your statement of the problem and you wrote [itex]S[/itex] as [itex]S(t_0, t)[/itex] suggesting that S is a function of an interval. This might give some reader a hint about the solution, but I omitted [itex]t_0[/itex] it since it isn't necessary.