Hi all,I am trying to numerically solve the Fokker-Plank equation for the variance in the Heston model using a grid of (x,v) using finite differences.x-> log return on stock v->varianceFor boundary conditions, I am referring to the paper- http://papers.ssrn.com/sol3/papers.cfm? ... id=1191962 which talks about boundary for p(x,v,t) at variance=0.There is another paper in which the author has used the grid of S,V ( Stock and Variance) instead of X,V that I am using. ( log returns and variance)http://papers.ssrn.com/sol3/papers.cfm? ... =1519674He has used boundaries p(s,v,t)=0 for s=0 and s=Smax. ( Page 6-7)My question is-How do I derive the boundary conditon for Vmax on the grid? Is it always necessary?Is there a standard approach on how boundary conditions are calculated?Or if I could you can point me to some existing paper on it, that would be great.Thanks in advance!

There are lots of general approaches for boundary conditions.For your particular question -- just do the stand-alone vol. problem and use the sameresult for that. Since V=infinity is a natural boundary (unreachable in finite time), just aboutanything will work. For example, p(v=vmax) = 0 or dp/dv(v=vmax)=0 for example, will both work fine,assuming vmax >> 1. The main problem with the literature on your question is that laughingly small values of vmax tend to be used.An easy way to achieve vmax >> 1 is to switch to new coords Y = V/(V+V0), then just lay a uniform grid i/N; i = 0,1,...,(N-1) Since the stand-alone vol. process has an exactly known transition density, plot your errors over your *entire grid*and you'll know if your solver is any good. (Ditto for the joint transition density)

The idea to use domain transformation is not widely known/used. Most models use ad-hoc truncation('5 std ...') And using transfomation makes it easier to find boundary conditions. A similar case is CIR when rmax ~ 2 !! Smax and stuff can result in energy leaks.I discuss some of these issues herehttp://papers.ssrn.com/sol3/papers.cfm?abstract_id=1552926 Here's a discussion that might be relevant as wellhttp://www.wilmott.com/messageview.cfm?catid=4 ... SGDBTABLE= QuoteIs there a standard approach on how boundary conditions are calculated?In general, ad hoc + heuristic, But one approach is Fichera theory. Search on this forum. And Feller's theory might be useful.A (marathon) discussion is herehttp://www.wilmott.com/messageview.cfm?catid=1 ... SGDBTABLE=

LEt me make a note that Fichera theory hasn't been sufficiently publicized (Daniel's books being exceptiona), especially not in connection with SDEs. A rare instance where this connection was made is the Friedman's "Stochastic Differential Equations and Applications I" pg. 206.

QuoteOriginally posted by: prosperoLEt me make a note that Fichera theory hasn't been sufficiently publicized (Daniel's books being exceptiona), especially not in connection with SDEs. A rare instance where this connection was made is the Friedman's "Stochastic Differential Equations and Applications I" pg. 206.I had a look at Friedman (btw good maths), 1976 edition pages 208-211.His use of Fichera Drift is the same as Fichera himself. The connection between Friiedman and Fichera on a bounded domain G seems to me to be this:a. If FD >= 0 then Friedman calls bdy(G) a 1-sided obstacle from the outside of G and in Fichera BC would correspond to Dirichlet BC.b. If FD <= 0 then Friedman calls bdy(G) a 1-sided obstacle from the inside of G and in Fichera BC would correspond to NO BC allowed/needed. (edit: changed to <=)Equation 4.6 is crucial.I think Only 2 pages to go on.In case a, Friedman states that a random process never enters G and in case b it never exits G. Thus, certain regions are nonattaiable.I think this would be a good issue to pursue from all perspectives as all the different bits of knowledge could be tied in together. What seems to receive little attentiion is well-posedness questions and energy inequalities to prove boundedness of PDE and FDM solutions.On page 267 Friedman discusses smooth domain transformation and states that FD is preserved under such a transformation. This is nice because I use this in ADE schemes e.g. y = S/(S + a). Alan uses it as well in NDSolve. Cheers//The book by Peter Kotelenz examines FD in the context of Stratonovich integral but I don't understand the deeper context. Of course, the monograph by Oleinik and Radkevic is a good source.