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Cuchulainn
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Partial Integro-Differential Equation (PIDE), Merton Jump Diffusion

December 29th, 2020, 7:51 pm

The standard PIDE (before it is subjected to the 'usual' independent variable transformation and domain  truncation) is

 (A) \[\frac{\partial V}{\partial \tau} = \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r-\lambda k) S \frac{\partial V}{\partial S} - (r+\lambda) V +\lambda\int_{0}^{\infty} V(S\eta)\Gamma_\delta(\eta)\,d\eta = 0, \]  

A big title and a big equation!

 There's a few FDM schemes that are variations on a theme 1) [$]x = log(S)[$] and [$]y = log(\eta)[$] and 2) truncation of both the PDE domain and the integral domain and 3) usual suspect boundary conditions. Everyone does this but is very crude.

My question now is: not using (1), (2), (3) but transforming across the board [$]S,\eta = (0,\infty)[$] to [$]x,y = (0,1)[$] by [$]x = S/(S+1)[$], [$]y = \eta/(\eta + 1)[$].

Now I go from a pide on [$](0,\infty)[$] to [$](0,1)[$] for both the PDE and IE terms. NO errors incurred.

Question; Is this a good starting point, numerically? Is there a (magical) reason why this might not work?

// A '42' question: why do most articles go for tricks (1), (2), (3) (saying it's handy is not an answer).  They work in practice but do they work in theory?
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl
 
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Alan
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Re: Partial Integro-Differential Equation (PIDE), Merton Jump Diffusion

January 5th, 2021, 7:50 pm

I don't think it's any better or worse than solving the PDE without the integral term by that same transformation. (Personally, it always seems like overkill to use PDE/PIDE numerics when there are analytic/quasi-analytic solutions). 
 
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Cuchulainn
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Re: Partial Integro-Differential Equation (PIDE), Merton Jump Diffusion

January 7th, 2021, 5:24 pm

I don't think it's any better or worse than solving the PDE without the integral term by that same transformation. (Personally, it always seems like overkill to use PDE/PIDE numerics when there are analytic/quasi-analytic solutions). 
That's almost a one-liner.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl