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Billy7
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Re: PDE methods for optimal quantizers of stochastic processes ? An example with Heston PDE.

March 7th, 2018, 6:44 pm

A while  collaborated with Alex Levin on a HW2-type PDE (no correlation); Long story short,   unable to get ADI to work and we went for Marchuck's 1-2-2-1 dimensional splitting method. Maybe someone at some stage could try it for Heston.
http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf
Can't imagine why ADI w/o mixed derivatives wouldn't work...what was the problem, you remember?
Interesting paper, but what is E in (2.2)? Comes out of thin air, or have I missed something? Also, it's not clear how it would be applied with mixed derivatives present as in Heston. But either way, if someone wants to try different efficient schemes, it should be on other models that don't have semi-analytical solutions (as we did in the paper), which are arguably better than Heston.
 
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Cuchulainn
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Re: PDE methods for optimal quantizers of stochastic processes ? An example with Heston PDE.

March 7th, 2018, 10:09 pm

A while  collaborated with Alex Levin on a HW2-type PDE (no correlation); Long story short,   unable to get ADI to work and we went for Marchuck's 1-2-2-1 dimensional splitting method. Maybe someone at some stage could try it for Heston.
http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf
Can't imagine why ADI w/o mixed derivatives wouldn't work...what was the problem, you remember?
Interesting paper, but what is E in (2.2)? Comes out of thin air, or have I missed something? Also, it's not clear how it would be applied with mixed derivatives present as in Heston. But either way, if someone wants to try different efficient schemes, it should be on other models that don't have semi-analytical solutions (as we did in the paper), which are arguably better than Heston.
The AL PDE had rho = 0 and ADI did not work. "E"  is the identity matrix (== I)in Russian.
Also, it's not clear how it would be applied with mixed derivatives 
The article does not discuss mixed derivatives.
 
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Billy7
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Re: PDE methods for optimal quantizers of stochastic processes ? An example with Heston PDE.

March 7th, 2018, 10:37 pm

Oh, should've guessed...
Well yes, since the original ADI is for rho=0 (no mixed derivatives), is unconditionally stable, so I was just wondering how it failed exactly.
Well if it is to be tried for Heston (or other SV models) as you suggested then one should know what to do with the mixed derivative, no? Does the original (Marchuk) article/book discuss examples of such splitting with mixed derivatives maybe?
 
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Cuchulainn
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Re: PDE methods for optimal quantizers of stochastic processes ? An example with Heston PDE.

March 8th, 2018, 9:16 am

Oh, should've guessed...
Well yes, since the original ADI is for rho=0 (no mixed derivatives), is unconditionally stable, so I was just wondering how it failed exactly.
Well if it is to be tried for Heston (or other SV models) as you suggested then one should know what to do with the mixed derivative, no? Does the original (Marchuk) article/book discuss examples of such splitting with mixed derivatives maybe?
AFAIR the boundary conditions and domain truncation were an issue.
Concerning Marchuk 1-2-2-1 _dimensional_ (x,y) splitting  it is indeed not clear how to split since [$]u_{xy}[$] has a foot in both camps as it were. Tbere is alo very little literature.

One alternative from Gil Strang (Strang-Marchuk splitting) is to split according to the PDE components 

A1: Pure elliptic part + convection
A2: Mixed derivatives
Separately, these are easy, haven't tried it. What do you think?

Some ideas
1. Is it possible to use all your insights into space discretisations and something else besides ADI for time? (e.g. Mathematica NDSolve or Boost odeint?)
2. How does non-zero [$]\rho[$] complicate thing? So, if is zero (just say) are things easier and where?
 
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Billy7
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Re: PDE methods for optimal quantizers of stochastic processes ? An example with Heston PDE.

March 8th, 2018, 2:19 pm

One alternative from Gil Strang (Strang-Marchuk splitting) is to split according to the PDE components 
A1: Pure elliptic part + convection
A2: Mixed derivatives
Separately, these are easy, haven't tried it. What do you think?
Yes, but that's what the ADI variants are doing (CS, HV), plus split A1 into two (for each direction). It's just that the mixed derivative part (A2) involves points in both directions, so needs to be solved explicitly...Not sure if the Marchuk scheme would still be 2nd order in time though. But in this case it would be pretty similar to the ADI variants anyway, so would it be preferable for some reason (more stable)? Don't know.
Some ideas
1. Is it possible to use all your insights into space discretisations and something else besides ADI for time? (e.g. Mathematica NDSolve or Boost odeint?)
2. How does non-zero [$]\rho[$] complicate thing? So, if is zero (just say) are things easier and where?
How about the BDF3 (or BDF2 if you don't like BDF3) as in the paper? It is faster than people think (in 2D and for time-independent coefficients), if you use a fast solver like those offered from within Eigen for example, as mentioned in the paper.
Wel if rho is zero then as you said, there's more literature and schemes are simpler, no? Apart from that can't think of anything else now.
 
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Cuchulainn
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Re: PDE methods for optimal quantizers of stochastic processes ? An example with Heston PDE.

March 8th, 2018, 2:47 pm

 It's just that the mixed derivative part (A2) involves points in both directions, so needs to be solved explicitly..

That's the crux. I hope to Strang Splitting with 2nd order implicit in each leg.

How about the BDF3 (or BDF2 if you don't like BDF3)
I haven't  looked at these to date :)

Not sure if the Marchuk scheme would still be 2nd order in time though.
I reckon it will break down to 1st order(?)
 
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Cuchulainn
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Re: PDE methods for optimal quantizers of stochastic processes ? An example with Heston PDE.

March 8th, 2018, 7:45 pm

How about the BDF3 (or BDF2 if you don't like BDF3) as in the paper? It is faster than people think (in 2D and for time-independent coefficients),

Billy,
Are you saying using the BDF instead of HV (MCS) for time discretisation while keeping the space discretisation as before? And BDF for each leg?

If yes, it is worth a shot.
(If no, why not?)