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Cuchulainn
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Fokker Planck PDE with delta impulse

September 24th, 2017, 4:44 pm

.I am not a fan of spreading out the delta. I like concentrating it on a node. The main reason is: you don't want to introduce an extraneous limit (besides taking the lattice spacing to zero). If you concentrate it on a node, it adjusts 'automatically' with the lattice spacing. For example with uniform spacing of a one-factor probem, my method has lattice Dirac delta with amplitude [$]1/\Delta x[$].
Alan

I have tried some of the other suspects but this one looks good (its integral is 1 and has compact support in a small interval). However, it is a rectangle and will that give nasty oscillations?
 
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Alan
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Re: Fokker Planck PDE with delta impulse

September 25th, 2017, 2:48 pm

I haven't seen that in 1D. 

In 2D, a problem that I have seen is that it is easy to get unwanted small negative values of [$]p(T,\vec{x}_T | \vec{x}_0)[$] far from the hotspot at [$]\vec{x}_0[$]. Whether or not these are oscillations, I'm not sure. This problem is fatal to maximum likelihood parameter estimation, my main interest at the moment.

Of course, any issue like this is scheme-and-model dependent. I am talking about mainly naive applications of NDSolve to popular stoch. vol. models, typically after some coordinate transformations.
 
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Billy7
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Re: Fokker Planck PDE with delta impulse

September 25th, 2017, 3:44 pm

I am suspecting that small negative values are not part of oscillations directly related to the initial discontinuity (since they're far from it?), but due to the diffusion-mixed derivative terms discretization (scheme not being monotone etc). Could be wrong though, I haven't looked at this equation for some time, but there are cures for that, how good they would prove to be in practice I don't know.