Some background to my Q.

Only closed form solutions lend lucid insight.

Lucid and incorrect go hand-in-hand in many cases! You need to quantify the assumptions in your formulas because closed solutions break down when parameters are outside some range. But only heuristics (aka a-posteriori hand-waving) tell us if they are good or not.

We have a large number of mathematically and numerically correct PDE/FDM schemes for a range equity and ir/fixed income (CIR, HW1, HW2) problems (CN,ADE, ADI, ADICS, MOL, Marchuk, Yanenko, Strang, Saulyev, DD exponential fitting.)

Concrete example: one 2d test is Kirk's (1995) approximation as in Haug (2007), pages 213-214. It is good in the main but deep OTM, ITM less so. But all the fdm schemes give the same answers as each other.

We are not use traditional FDM, FFT or MC. Too much hassle.

Similar excellent results HW1, HW2 incorporating instantaneous forward rate.

Next step: a few of us to apply HW2 to CMS spread range accruals.

*Need to define the scope first..*
Hypothesis: the best approximation to a closed solution is to apply [3,7] robust FD schemes to the PDE, letting dt -> 0, h -> 0 and checking results are the same. They all can't be wrong because each one is both a generalist and a specialist.