Friends would recall when I took the Fokker-planck solution and "backed out" standard normal variance increments between any two nodes at time tn and time tn+1. We were excited when we could accurately apply Ito change of variable formula using those Zt increments. But other SDEs were not working any more with those transition Zt increments and other dt-integrals were losing variance. Then we constructed an integrated Z(tn) from transition standard variance increments using transition probabilities. I was comparing variance of these reconstructed Z(tn) with variance of integrated values of standard normals used in monte carlo simulations but I was seriously puzzled that variance of reconstructed Z(tn) on fokker-planck grid using transition probabilities was far lesser than integrated values of normal constructed from integration of driving standard normals in monte carlo. Finally I blamed these discrepancies on inaccuracies of the Fokker-planck based transition probabilities grid and started an independent standard normal increments based grid.difference with fokker-planck solution is that Fokker-planck solution automatically calculates any change in variance due to drift while other monte carlo-like(with random numbers replaced with stochastic integrals) would have to explicitly account for any contraction due to drift and would have to make extra calculations for that.

Though there were indeed small inaccuracies on fokker-planck based transition probabilities grid but now it makes sense that underlying driving normal distribution had lesse variance because it contracted due to mean reverting drift and our backed out transition normals were quite close to accurate values and therefore Ito change of variable for that particular SDE was working very well. In fact due to variable local contraction due to drift, our retrieved "normal"(with integration of transition normal on fokker-planck based transition probabilities grid) was not even a perfectly straight line. However transition normal increments everywhere had unit variance(approximately different due to slight lack of accuracy).

No wonder that we could not integrate any other SDE on the grid of our retrieved "normal"(with integration of transition normal on fokker-planck based transition probabilities grid). And we also noticed that dt-integrals were fast losing variance on this Z-grid which again makes sense since the constructed normal grid had contraction due to drift.

And now it all seems to make sense since Fokker-planck algorithm automatically calculates local adjustment in variance of underlying integrated normal grid due to mean-reverting and variance changing drift.

I am sitting back for next 2-3 days brainstorming and thinking of the best way to go ahead on the transition probabilities project since a lot of things have dramatically changed due to better understanding of underlying phenomenon.

Again our roadmap of research is the same. I want to complete the transition probabilities for Bessel SDEs part in a very good way and then move to caclulation of dt-integrals and dz-integrals(possibly on different sister grids) with transition probabilities. I think we would be able to solve general(not necessarily bessel form) SDEs which are counterparts of their bessel form on the same grids since underlying Z-grid remains the same.

And then we would move to monte carlo and transition probabilities grid generation of more general SDEs where each term would possibly include time functions(exponentials and powers of time and others) multiplying algebraic functions of SDE variable. So I hope that we would continue to do some exciting work in coming months and then move on to inference, parameter estimation, filtering, AI and other interesting things.