Intuition of the Kolmogorov Equations
I will try to answer to your last question. I did not read the paper but in models with higher dimensions, it is always difficult to find an analytical solution. 
If there exists an analytical solution (for a very basic model with a one-state variable), it is possible to derive the initial conditions for your control and state variable from your differential equations.

However, in systems where there does not exist an analytical solution, you are supposed to solve it numerically, in which case you must give a numerical value for one of your variables (or more until you find steady state values for all your variables.)

After, you can find the steady-state values of your variables in which case, your initial conditions should not be so far from the steady state level (otherwise, there would be some convergence problems in your model if you don't choose the appropriate initial values, close to steady state. It is another issue.)