Fichera in 'Unconditionally Stable+ 2nd Order......

Fichera in 'Unconditionally Stable+ 2nd Order......

Postby paddy » Thu May 02, 2013 10:21 am

Daniel,



I have finished reading your paper 'Unconditionally Stable & Second order ...' at SSRN. Thank you it was a great read. I particularly liked the section on gridding - very useful stuff.



I have a question regarding the Fichera theory. I had not heard of Fichera prior to reading this and it looks very useful. I have a question regarding your definitions however, which I cannot look up because I don't have access to Fichera's originals. When I apply the definitions given in this paper all of the signs in well known inequalities (e.g. Feller's inequality) seem backwards. Please let me know what I am doing wrong.



Take for instance the Feller process



dx = k*(r-x)*dt + s*sqrt(x)*dW



with Fokker-Planck Equation (with subscript notation for derivatives)



f_{t} = 0.5*x*s^2*f_{xx} + [s - k*(r-x)]*f_{x} + k*f,



which is in the form of Equation (22) in your paper with operator



Lf = 0.5*x*s^2*f_{xx} + [s - k*(r-x)]*f_{x} + k*f.



The two functions for Equation (20) (i.e. for the Fichera function) are



a = 0.5*x*s^2



and



b = [s - k*(r-x)],



so that the Fichera function is



Fichera = [0.5*s^2 - k*(r-x)]v



where v is the outward normal. Now at x=0 the inward normal is v=1

and so



Fichera = [0.5*s^2 - kr].



No BCs are allowed at zero when this is positive, which gives



2*k*r < s^2



which is of course the wrong way round!!! Are my readings of the definition of the operator L wrong?



Thank you for your time - I learn't a lot from this article.



Paddy
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Postby Cuchulainn » Sun May 12, 2013 3:47 pm

Thanks, paddy for the feedback.



Some discussions can be found here; it seems like an open issue



http://www.wilmott.com/messageview.cfm? ... adid=94341
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Re: Fichera in 'Unconditionally Stable+ 2nd Order......

Postby Cuchulainn » Wed May 07, 2014 9:56 pm

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