Page **21** of **23**

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Posted: **April 6th, 2019, 6:02 pm**

by **JohnLeM**

OK, so just so that I get it straight:

1. call payoff (S - K)+ is **not **BV and there is NO sampling sequence for it which converges faster than 1/N

2. binary payoff Heaviside(S - K) is BV and there IS a sampling sequence for it which converges faster than 1/N

Correct?

Nope: Correct for binary (thank, that was the name I confused with barrier option!). But, as call Payoff (x-K)^+ have second order derivatives that are measure valued, there exists for any N, any probability measure [$]\mu[$] (with technical assumptions here) a sampling sequence [$]x^1,\ldots,x^N[$]such that

[$] | \int_{R^D} (x_d-K)^+ d\mu(x) - \frac{1}{N} \sum_{n=1}^N (x_d^n-K)^+ | \le \frac{C}{N^2} [$]
for any strike K, d= 1..D.
More precisely, without proving this result, one can not pretend to break the curse of dimensionality, would he use deep tralala networks or not.

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Posted: **April 6th, 2019, 6:37 pm**

by **Rayavidas**

Really interesting that file Kernel for deep learning

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Posted: **April 6th, 2019, 6:43 pm**

by **ISayMoo**

Then I'm still confused. Can you clarify the relation between BV functions, convergence rates and derivative payoffs?

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Posted: **April 7th, 2019, 9:01 am**

by **JohnLeM**

Then I'm still confused. Can you clarify the relation between BV functions, convergence rates and derivative payoffs?

ISayMoo, you are not confused at all, you are clearly sharp minded. This is a very pertinent question, it is exactly the same question that the one you asked concerning the constant above. Hence my answer is the same one: we are currently releasing this part of our research, this is the paper I owe to Cuchulain. But you'll have to wait a little bit for the final paper, because we cannot find neither the time, nor the money, to do this job.

Alternatively, wait for the AI community or ask to banks quantitative research groups : with 2 billions public French euros, and the full power of private industry fundings, they could maybe deign to justify their wages with the help of a cat ?

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Posted: **April 7th, 2019, 9:55 am**

by **Cuchulainn**

Excuses

Function of BV are a RED HERRING when talking about payoff continuity. They are almost useless AFAIK. The last time someone mentioned BV to me was way back in the mid 20th century in Lebesgue integration class.

It is a well-known problem in PDE/FDM finance.

This article first pin-pointed the issue for Black Scholes. The problem was glossed over or ignored (e.g. my questions at a conference LOL) till then

https://wwwf.imperial.ac.uk/~ajacquie/I ... uffyCN.pdf

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Posted: **April 7th, 2019, 10:21 am**

by **Cuchulainn**

Here is a seminal paper on PDE with discontinuous payoff (initial condition). This was known in our group by 1974 (it's in the FEM book by Gil Strang and George Fix).

https://wwwf.imperial.ac.uk/~ajacquie/I ... uffyCN.pdf
So, I'm wondering what new revelations to expect in the present discourse..

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Posted: **April 7th, 2019, 11:08 am**

by **JohnLeM**

Here is a seminal paper on PDE with discontinuous payoff (initial condition). This was known in our group by 1974 (it's in the FEM book by Gil Strang and George Fix).

https://wwwf.imperial.ac.uk/~ajacquie/I ... uffyCN.pdf
So, I'm wondering what new revelations to expect in the present discourse..

This link points to your 2004 article concerning convexity problems of CN methods. Is it the link you wanted to put ?

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Posted: **April 7th, 2019, 11:22 am**

by **Cuchulainn**

Here is a seminal paper on PDE with discontinuous payoff (initial condition). This was known in our group by 1974 (it's in the FEM book by Gil Strang and George Fix).

https://wwwf.imperial.ac.uk/~ajacquie/I ... uffyCN.pdf
So, I'm wondering what new revelations to expect in the present discourse..

**This link points to your 2004 article concerning a) convexity problems of CN methods. b) Is it the link you wanted to put ?**

a) No, b) YES.

But I don't know what you mean by 'convexity' here.

// In short, A-stable methods versus L-stable methods.

This is like pulling teeth.

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Posted: **April 7th, 2019, 11:38 am**

by **JohnLeM**

Here is a seminal paper on PDE with discontinuous payoff (initial condition). This was known in our group by 1974 (it's in the FEM book by Gil Strang and George Fix).

https://wwwf.imperial.ac.uk/~ajacquie/I ... uffyCN.pdf
So, I'm wondering what new revelations to expect in the present discourse..

**This link points to your 2004 article concerning convexity problems of CN methods. Is it the link you wanted to put ?**

No, YES.

But I don't know what you mean by 'convexity' here.

// In short, A-stable methods versus L-stable methods.

Well, I refer to convexity problems because, as you noticed in your paper, CN schemes produce oscillations while computing second-order derivatives, due to complex valued eigenvalues. Can't monotone schemes solve the issue ? Indeed, it seems that your exponential fitting does the job, isn't it ?

[added] I see, this is probably what you are meaning : "A stable" versus "L stable"...

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Posted: **April 7th, 2019, 11:48 am**

by **JohnLeM**

Excuses

You're right, I should finish this paper right now.

**Cuchulainn:**
Function of BV are a RED HERRING when talking about payoff continuity. They are almost useless AFAIK. The last time someone mentioned BV to me was way back in the mid 20th century in Lebesgue integration class.

Well..binary options are BV functions. Calls are functions which derivatives are BV. This functional space is the most used one in the Finance industry to model payoffs...

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Posted: **April 7th, 2019, 12:03 pm**

by **Cuchulainn**

**This link points to your 2004 article concerning convexity problems of CN methods. Is it the link you wanted to put ?**

No, YES.

But I don't know what you mean by 'convexity' here.

// In short, A-stable methods versus L-stable methods.

Well, I refer to convexity problems because, as you noticed in your paper, CN scheme produces oscillations while computing second-order derivatives, due to complex valued eigenvalues. Can't monotone schemes solve the issue ? Indeed, it seems that your exponential fitting does the job, isn't it ?

[added] I see, this is probably what you are meaning : "A stable" versus "L stable"...

2 orthogonal issues

Exponential fitting resolves convection dominance, resulting in a monotone scheme.

Fully implicit is stable.

CN oscillate in [$]V, \Delta, \Gamma[$].

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Posted: **April 7th, 2019, 12:16 pm**

by **Cuchulainn**

**This link points to your 2004 article concerning convexity problems of CN methods. Is it the link you wanted to put ?**

No, YES.

But I don't know what you mean by 'convexity' here.

// In short, A-stable methods versus L-stable methods.

Well, I refer to convexity problems because, as you noticed in your paper, CN schemes produce oscillations while computing second-order derivatives, due to complex valued eigenvalues. Can't monotone schemes solve the issue ? Indeed, it seems that your exponential fitting does the job, isn't it ?

[added] I see, this is probably what you are meaning : "A stable" versus "L stable"...

Ah, nowhere in my paper do I mention convexity(is that your gamma??)

See also, my previous post.

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Posted: **April 7th, 2019, 12:27 pm**

by **JohnLeM**

Ah, nowhere in my paper do I mention convexity(is that your gamma??)

See also, my previous post.

convexity is my very personal way to refer to these problems of oscillations. But even if there exists a more appropriate wording for that, I will refer now to these problems as deep neural convexity problem to get granted !

Cuchullain, speaking of FD schemes, have you

read this very excellent good article ? More seriously, I just wanted to point out that one can very well define non local operators while designing finite difference schemes.

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Posted: **April 7th, 2019, 12:42 pm**

by **Cuchulainn**

*convexity is my very personal way to refer to these problems of oscillations. *

I hope you use standardised notation in your new article. BTW are you a mathematician./CS?engineer/physicist by training?

BTW there be no 'h' in Nicolson..

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Posted: **April 7th, 2019, 12:46 pm**

by **JohnLeM**

*convexity is my very personal way to refer to these problems of oscillations. *

I hope you use standardised notation in your new article. BTW are you a mathematician./CS?engineer/physicist by training?

I will do my very best, because I know that you will kick my ass if I don't. I had my applied math Ph-D from Bordeaux University in 1996.