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Cuchulainn
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Hunting the Dirac Delta

I am looking for a good approximation ... there are many out there but ideally it should be

1. Ideally, $C^\infty$ smooth (a tempered distribution).
2. Always positive
3. Preserves/compatible with monotone schemes
4, Works with the PDE for gamma sensitivity
5. A bunch of deltas out there that don't cut the mustard.
6, No handwaving and well-known fixes.

The derivative of Fermi Dirac function looks not bad but I can't see the full consequences. An approximation of an approximation makes things worse?
They sought it with thimbles, they sought it with care; They pursued it with forks and hope; They threatened its life with a railway-share; They charmed it with smiles and soap.

Alan
Posts: 2751
Joined: December 19th, 2001, 4:01 am
Location: California
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Re: Hunting the Dirac Delta

Suppose a lattice $\{x_i\}$ with spacing $dx$. Assume a class of approximate Dirac Deltas functions with area=1 for all and centered at  $x_i = 0$. I think the critical property is that, evaluated at the centered lattice point, the function value tends to $1/dx$ as $dx \rightarrow 0$.
Once you've got that, I will guess just about anything will work.

I always use/suggest the non-smooth function $f(x) = \frac{1}{dx} 1_{\{|x| < 0.5 \, dx\}}$.
Sure, it violates 1. If it's really important to have Property 1, you could always just round the corners.

Alan
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Joined: December 19th, 2001, 4:01 am
Location: California
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Re: Hunting the Dirac Delta

Cuchulainn
Topic Author
Posts: 18490
Joined: July 16th, 2004, 7:38 am
Location: Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch

Re: Hunting the Dirac Delta

Suppose a lattice $\{x_i\}$ with spacing $dx$. Assume a class of approximate Dirac Deltas functions with area=1 for all and centered at  $x_i = 0$. I think the critical property is that, evaluated at the centered lattice point, the function value tends to $1/dx$ as $dx \rightarrow 0$.
Once you've got that, I will guess just about anything will work.

I always use/suggest the non-smooth function $f(x) = \frac{1}{dx} 1_{\{|x| < 0.5 \, dx\}}$.
Sure, it violates 1. If it's really important to have Property 1, you could always just round the corners.
Unfortunately, this option leads to quite a lot of numerical issues and the inevitable fixes in the literature, which is a bit depressing.
Taking the basic SDE dX = sig dW we get the 1-factor FPE diffusion PDE Q_t = 1/2 sig^2 Q_xx  with delta d(x) as initial condition. Even for this special case of d(x) = 1/h (h = dx):

1. Crank Nicolson will give oscillation or negative values unless dt == k < O(h^2). So, it behaves like explicit Euler.
2. Other "suspects" fdm also have issues.
3. A solution is to transform  to natural time of Brownian motion $\tau = \sqrt t$ which fixes this particular problem, giving positive density under all combinations.

At this stage, is there a way to replace d(x) in the initial condition and "hide" it somewhere else in the PDE? Having it elsewhere maybe less damaging that as a spike?
They sought it with thimbles, they sought it with care; They pursued it with forks and hope; They threatened its life with a railway-share; They charmed it with smiles and soap.

Cuchulainn
Topic Author
Posts: 18490
Joined: July 16th, 2004, 7:38 am
Location: Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch

Re: Hunting the Dirac Delta

Yes, that's an idea, the bump functions, maybe. They are smooth but not $C^{\infty}$ .. don't know what the relation is with delta functions. They do look promising.
They sought it with thimbles, they sought it with care; They pursued it with forks and hope; They threatened its life with a railway-share; They charmed it with smiles and soap.

Cuchulainn
Topic Author
Posts: 18490
Joined: July 16th, 2004, 7:38 am
Location: Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch

Re: Hunting the Dirac Delta

Yes, that's an idea, the bump functions, maybe. They are smooth but not $C^{\infty}$ .. don't know what the relation is with delta functions. They do look promising.
Aka mollifiers in Functional Analysis. They must have integral == 1, which is tricky, especially in  higher dimensions.
They sought it with thimbles, they sought it with care; They pursued it with forks and hope; They threatened its life with a railway-share; They charmed it with smiles and soap.

Alan
Posts: 2751
Joined: December 19th, 2001, 4:01 am
Location: California
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Re: Hunting the Dirac Delta

There is an example of using my non-smooth suggestion with NDSolve (including code) in my Vol II book, pages 462-464. It works quite well, as Fig 10.5 there shows.

Cuchulainn
Topic Author
Posts: 18490
Joined: July 16th, 2004, 7:38 am
Location: Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch

Re: Hunting the Dirac Delta

There is an example of using my non-smooth suggestion with NDSolve (including code) in my Vol II book, pages 462-464. It works quite well, as Fig 10.5 there shows.
This looks very good, Alan.
I reckom NDSolve (using Bulisch-Stoer) can handle many kinds of pesky deltas:

1.  avoiding oscillations and negative values
2. what is accuracy order (high?), order 4 in space, order 5 in time?
3. sanity check: integral == 1. (cdf)
4. what about small and large T?

Now, I have modelled the PDE for option gamma (differentiate  BSPDE twice wrt S) with a Gaussian as delta. I used C++, centred differencing in S and Bulisch-Stoer in time (Boost C++ odeint library). I get excellent results.

Will post details later today.

Meanwhile, back at the ranch, people are looking for fixes for traditional fdm. It is based on flawed assumptions and expectations. A kind of wasted effort.
They sought it with thimbles, they sought it with care; They pursued it with forks and hope; They threatened its life with a railway-share; They charmed it with smiles and soap.

Cuchulainn
Topic Author
Posts: 18490
Joined: July 16th, 2004, 7:38 am
Location: Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch

Re: Hunting the Dirac Delta

Solving the PDE for gamma using Bulisch-Stoer and Cash Karp with Gaussian delta function
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They sought it with thimbles, they sought it with care; They pursued it with forks and hope; They threatened its life with a railway-share; They charmed it with smiles and soap.

Cuchulainn
Topic Author
Posts: 18490
Joined: July 16th, 2004, 7:38 am
Location: Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch

Re: Hunting the Dirac Delta

We also stress tested for Speed = dgamma/dS . We used the Complex Step Method to compute the derivative of delta, which is Speed’s payoff.

// Distributions and delta functions
using cvalue_type = std::complex<value_type>;
cvalue_type deltaFunction(cvalue_type z, value_type a, value_type eps)
{
return std::exp(-(z - a)*(z - a) / (2.0*eps*eps))
/ std::sqrt(2.0*3.14159265359*eps*eps);
}

value_type deltaFunctionReal(value_type x, value_type a, value_type eps)
{
return std::exp(-(x - a)*(x - a) / (2.0*eps*eps))
/ std::sqrt(2.0*3.14159265359*eps*eps);
}

value_type derivativeDeltaFunction(value_type x, value_type a, value_type eps,
value_type h)
{
//     value_type h = 0.01;
cvalue_type z(x, h);
return std::imag(deltaFunction(z, a, eps))/h;
}

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They sought it with thimbles, they sought it with care; They pursued it with forks and hope; They threatened its life with a railway-share; They charmed it with smiles and soap.

Alan
Posts: 2751
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Re: Hunting the Dirac Delta

There is an example of using my non-smooth suggestion with NDSolve (including code) in my Vol II book, pages 462-464. It works quite well, as Fig 10.5 there shows.
This looks very good, Alan.
I reckom NDSolve (using Bulisch-Stoer) can handle many kinds of pesky deltas:

1.  avoiding oscillations and negative values
2. what is accuracy order (high?), order 4 in space, order 5 in time?
3. sanity check: integral == 1. (cdf)
4. what about small and large T?

Now, I have modelled the PDE for option gamma (differentiate  BSPDE twice wrt S) with a Gaussian as delta. I used C++, centred differencing in S and Bulisch-Stoer in time (Boost C++ odeint library). I get excellent results.

Will post details later today.

Meanwhile, back at the ranch, people are looking for fixes for traditional fdm. It is based on flawed assumptions and expectations. A kind of wasted effort.
NDSolve is order 4 in space and uses adaptive time-stepping. In my experience it handles both small and large T well. It's especially good for large T when the solution tends to a constant-in-time function (the soln to the associated elliptic problem), as it will take *very* large time steps when warranted.

Cuchulainn
Topic Author
Posts: 18490
Joined: July 16th, 2004, 7:38 am
Location: Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch

Re: Hunting the Dirac Delta

The "box" delta you use is also very good. I used it with the Gamma PDE and it works fine.

I also did an initial heat kernel with ADE with OK results.  CN is all over the place.
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They sought it with thimbles, they sought it with care; They pursued it with forks and hope; They threatened its life with a railway-share; They charmed it with smiles and soap.

Cuchulainn
Topic Author
Posts: 18490
Joined: July 16th, 2004, 7:38 am
Location: Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch

Re: Hunting the Dirac Delta

Can FPE be written in the self-adjoint form?

$\frac{\partial p}{\partial t} =$ (A(x) (B(x) p)_x)_x ?

and for n-factor.
They sought it with thimbles, they sought it with care; They pursued it with forks and hope; They threatened its life with a railway-share; They charmed it with smiles and soap.

katastrofa
Posts: 7133
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

Re: Hunting the Dirac Delta

You need delta or actually its derivative?

Cuchulainn
Topic Author
Posts: 18490
Joined: July 16th, 2004, 7:38 am
Location: Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch

Re: Hunting the Dirac Delta

You need delta or actually its derivative?
I have now no issues with delta nor its derivative. BTW p is transition probability, p(t = 0) = delta.

the Q can I write FPE in verry nice compact form and what are the advantages

p_t = (A (B p_x))_x

it's just another PDE.
They sought it with thimbles, they sought it with care; They pursued it with forks and hope; They threatened its life with a railway-share; They charmed it with smiles and soap.