Sorry, I was too sleepy last night to express myself clearly and I was being very incoherent. But the true idea that I had yesterday was that we should freeze the expectations made at earlier times and simply forward transport them along the Z-standard deviation grid associated with standard normal probability mass. By Z-standard deviation grid, I mean an expanding or contracting grid where grid cells are divided so that probability mass within each cell remains constant over time. So if there are payoffs at t1, t2, t3 and t4. We could freeze the expected payoffs at t1, on expanding/contracting standard deviations grid that preserves the probability mass in them and simply transport the expectation at t1 to respective cells on standard deviations grid at time t2 if we need to analyze the probability distribution at time t2 otherwise we could simply transport the frozen expectations to corresponding standard deviations grid at terminal time t4.
This could work for many types of "payoffs". But it still remains to be seen if the idea would work for all "payoffs". But I am sure exploring it will help solve the most general problem.
I am trying to superimpose a standard deviations grid on regular fixed grid and trying to learn "calculus" to convert expectations across two grids.