Green's function for Parabolic PDEs

Green's function for Parabolic PDEs

Postby Taras » Tue Apr 24, 2007 1:35 pm

Hello everyone,



I suppose that this is more of a PDE question, but ultimately some numerical analysis will be used.



I have a 2nd order pde on my hands for a density in stock and interest rate in time. I need to step through this pde forward and update one of the coefficients.



The initial density is a delta function and I tried anothe transform so that I have a normal initially, but it is still too thin and steep. So, finite differences are difficult. I would need to integrate over this density, so I would need some specialized mesh.



So, I am turning to Green's functions and I will try to get a closed form. Does anyone know of a good book with many examples of Green's functions, especially applicable to finance? Or, if there is some work that has been done with Green's functions for 2nd order pdes (with 2 or more state variables), please let me know, so that I do not reinvent the wheel :)



Thanks,



Taras
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Postby Cuchulainn » Thu Apr 26, 2007 10:21 pm

Hi Taras,

I think that Alan Lewis might be a good source. Do you know his books?



Do you have a link URL to the work?
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Numerics for cross derivative term in PDE

Postby Taras » Mon May 14, 2007 1:21 pm

Hi Daniel,



For some reason I did not see you reply until today. And I see that you replied quite a while ago, sorry.



Actually, for what I am working on, PDE for the value of a derivative in a diffusion for stock and Vasicek interest rates, there is a paper by Deakin and Mallier (can be googled) on Green's function for the constant coefficient case. There are some bugs in the paper, but the authors are working on a revision and I have corrected some (or hopefully all) myself. In my case, I have non-constant coefficients, but if I look at small time step, I can pretend constant.



But, as a follow-up, a general question on a parabolic PDE with a cross derivative term. For numerical solutions we want to not have this term (e.g., do a transform) for finite differences or something else. Is there an approach for a solution that you would recommend? Is there a good source for reading about different options for handling the cross derivative term? All texts that I have seen avoid talking about it like a plague.



I have not looked at you FDM book yet.



Thanks,

Taras
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Postby Cuchulainn » Fri May 18, 2007 7:18 pm

Taras,

R. Sheppard has done an excellent piece of work in this area. He has 4th and 5th order covergence using



- exponential fitting

- Soviet splitting

- extrapolated Euler (thus, NO Crank Nicolson)

- Yanenko's scheme for mixed derivatives



He has proved convergence/stability + extensive numerical tests



http://www.datasimfinancial.com/frm/viewtopic.php?t=101
Last edited by Cuchulainn on Sun Sep 02, 2007 2:50 am, edited 2 times in total.
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Mixed Derivatives with FDM schemes

Postby Taras » Mon May 21, 2007 12:12 pm

Thank you for the reply. I am looking into his work.



In different texts and papers, it is mentioned that the mixed derivative is problematic, but the reason is not explained. Page 6 of the attached paper talks about a desired form (strictly diagonally dominant with positive diagonal and non-positive off-diagonal elements) of a matrix needed for stability reasons. Discretizations with mixed derivatives do not lead to such a matrix. Is this the issue at hand or is there more that you can tell me?



I also just got your book, so I will look through. In it, I see that you have Yanenko's scheme.



What is your experience with Craig-Sneyd? It is not one of the methods listed, but I saw that you teach it in your course.



Thanks,

Taras



P.S. I may be chiming in with C# questions/comments during the summer.
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Operator Splitting - Ikonen and Toivanen.pdf
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Postby Cuchulainn » Mon May 21, 2007 2:15 pm

Taras,

Here is a discussion precisely on the points you ask about. Hope this helps.



http://www.wilmott.com/messageview.cfm? ... adid=49248



In he FDM book we use Janenko's approach for mixed derivatives.



Craig-Sneyd is in fact a splitting method. I have not used it but it seems to be good.
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Postby Taras » Mon May 21, 2007 3:43 pm

OK, I see. For a FDM scheme we want to have a non-singular M-matrix so that we have a monotonic scheme. The presence of mixed derivatives will not (in general) give us the M-matrix we want.



I will look into this more.



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Postby prospero » Wed Jan 16, 2008 10:44 am

Some interesting work can also be found here

http://www.win.ua.ac.be/~kihout/pub.html



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Postby Cuchulainn » Thu Jan 17, 2008 2:22 pm

This is an interesting article. Looking at equation (2.1) we have no correlation between r and the other terms, so the Sheppard approach could be applied in a splitting framework. Has anyone done this?



A question on boundary conditions: if I wish to solve 2.8 numerically, can we use the same BConditions as one would use in CIR; Do we need to take the Feller condition into consideration?
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Hello

Postby davidpeter » Mon Jan 19, 2009 12:32 pm

Is this the issue more that you can tell me?Is there a good source for reading about different options for handling the cross derivative term? All texts that I have seen avoid talking about it but if I look at small time step.
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Postby Cuchulainn » Wed Jan 21, 2009 3:12 pm

The answer is given here



http://www.datasimfinancial.com/forum/v ... .php?t=101



and in the thesis of Roelof Sheppard
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