I am approximating a PDE whose underlying mean reverting lognormal process has drift and volatility depending on the process realization. (I kind of the CIR stochastic process)When I use the fully implicit method, this is unstable for some values of the mean reverting speed.Since a fully implicit method is "unconditionally stable", is the instability possible in my framework or I am making mistakes in my matlab code?

Solved... thanks, by the way

Also space discretization can cause the discretization to be unstable.A sufficient condition for the stability is that the matrix resulting from the discretization has M-matrix property.A matrix has M-matrix property if1) it has positive diagonal,2) it is diagonally dominant (on each row, the diagonal is larger than the sum of absolute values of off diagonal entries),3) off diagonal entries are non positive.Typically discretizations of first-order derivatives and second-order cross derivatives can lead to positive off diagonal entries.

Thank you.My problem was the "ln(r)" in the drift which gave me negative values for 0<r<1 involving instability.

QuoteOriginally posted by: TeneAlso space discretization can cause the discretization to be unstable.A sufficient condition for the stability is that the matrix resulting from the discretization has M-matrix property.A matrix has M-matrix property if1) it has positive diagonal,2) it is diagonally dominant (on each row, the diagonal is larger than the sum of absolute values of off diagonal entries),3) off diagonal entries are non positive.Typically discretizations of first-order derivatives and second-order cross derivatives can lead to positive off diagonal entries.The combination of convection-diffusion terms can lead to spatial instability. This can be resolved by exponential fitting,resulting in an M-matruix.Some convoluted meshing may help mixed derivatives, again resulting in an M-matrix but AFAIK a non-constant mesh is needed. Using Yanenko resolves these problems.

QuoteThe combination of convection-diffusion terms can lead to spatial instability. This can be resolved by exponential fitting,resulting in an M-matruix. Some convoluted meshing may help mixed derivatives, again resulting in an M-matrix but AFAIK a non-constant mesh is needed. Using Yanenko resolves these problems.Could you suggest me some references about exponential fitting and Yanenko scheme?Thanks

QuoteOriginally posted by: Gerry105QuoteThe combination of convection-diffusion terms can lead to spatial instability. This can be resolved by exponential fitting,resulting in an M-matruix. Some convoluted meshing may help mixed derivatives, again resulting in an M-matrix but AFAIK a non-constant mesh is needed. Using Yanenko resolves these problems.Could you suggest me some references about exponential fitting and Yanenko scheme?ThanksSure.See the thesis by Roelof Sheppard that says it all.http://www.datasimfinancial.com/forum/v ... .php?t=101 And for foundations of expo fittinghttp://www.datasimfinancial.com/UserFiles/arti ... iel3.pdfMy PDE/FDM book also discusses these topics.

I've just glanced at the works you gave me... they are really what I was looking for.I hope to solve my problem.Thank you again, I'll let you know.

QuoteOriginally posted by: Gerry105I've just glanced at the works you gave me... they are really what I was looking for.I hope to solve my problem.Thank you again, I'll let you know.You're welcome.Regarding Yanenko, the FD2 code here FD2 uses it and you might get some hints there (although I use ADE rather than Soviet Splitting in this case, so it is a bit off-topic regarding OP I suppose).

Thanks again, you really helped me with these references. I solved the problem with the exponential fitting. Now the FD method is stable for all the parameter values (even with a sigma close to zero).I use the work of Daniel J. Duffy, a very helpful reference.As soon as I have time, I will glance at Yanenko and your book, too. Thanks againGerardo

That was quick coding