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by JohnLeM
April 8th, 2019, 12:39 pm
Forum: Numerical Methods Forum
Topic: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?
Replies: 327
Views: 12531

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

You're welcome. BTW I see that Jim Douglas Jr. passed  few years ago. https://sinews.siam.org/Details-Page/Obituaries-Jim-Douglas-Jr Peter Lax is still here. Jim Douglas is the one to be credited for ADI methods ? I did not know that either. Thanks again. Concerning Lax, to me he is a little bit sc...
by JohnLeM
April 8th, 2019, 11:43 am
Forum: Numerical Methods Forum
Topic: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?
Replies: 327
Views: 12531

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

You are not alone! Numerical analysts know these things, https://upload.wikimedia.org/wikipedia/en/thumb/9/98/PhyllisNicolson.jpg/220px-PhyllisNicolson.jpg Ooops... I used CN schemes for years believing that Nicolson was a man ! These schemes are one of the most used schemes in numerical analysis, ...
by JohnLeM
April 7th, 2019, 12:55 pm
Forum: Numerical Methods Forum
Topic: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?
Replies: 327
Views: 12531

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

BTW there be no 'h' in Nicolson..
Isn't he this guy ? (kidding, there is no h, you are right).
Image
by JohnLeM
April 7th, 2019, 12:46 pm
Forum: Numerical Methods Forum
Topic: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?
Replies: 327
Views: 12531

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

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...
by JohnLeM
April 7th, 2019, 12:27 pm
Forum: Numerical Methods Forum
Topic: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?
Replies: 327
Views: 12531

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

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 pr...
by JohnLeM
April 7th, 2019, 11:48 am
Forum: Numerical Methods Forum
Topic: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?
Replies: 327
Views: 12531

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

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 ...
by JohnLeM
April 7th, 2019, 11:38 am
Forum: Numerical Methods Forum
Topic: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?
Replies: 327
Views: 12531

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

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/IC_Num_Methods/IC_Num_Methods_Docs/Literature/DuffyCN.pdf So, I'm wondering what new revela...
by JohnLeM
April 7th, 2019, 11:08 am
Forum: Numerical Methods Forum
Topic: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?
Replies: 327
Views: 12531

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

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/IC_Num_Methods/IC_Num_Methods_Docs/Literature/DuffyCN.pdf So, I'm wondering what new revela...
by JohnLeM
April 7th, 2019, 9:01 am
Forum: Numerical Methods Forum
Topic: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?
Replies: 327
Views: 12531

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

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...
by JohnLeM
April 6th, 2019, 6:02 pm
Forum: Numerical Methods Forum
Topic: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?
Replies: 327
Views: 12531

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

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,...
by JohnLeM
April 6th, 2019, 12:37 pm
Forum: Numerical Methods Forum
Topic: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?
Replies: 327
Views: 12531

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

>> And why is this train of thought relevant here? e.g. why do you want to differentiate a payoff. I was trying to explain to @ISayMoo that Calls are somehow one derivative smoothers than Autocalls. Hence a sampling method, meaning a Monte-Carlo like method of kind [$] \int_{R^D} P(x) d\mu(x) \sim \...
by JohnLeM
April 6th, 2019, 11:32 am
Forum: Numerical Methods Forum
Topic: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?
Replies: 327
Views: 12531

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

Fair enough. Aka indicator/ Kronecker function. https://en.wikipedia.org/wiki/Dirac_measure The question is still: "the derivative of a call option is a barrier option". Is that what you mean? (x-K)^+ ' = {0, x< K, 1, x> K}. Can't I call this a barrier option ? I only see the derivative of a call p...
by JohnLeM
April 6th, 2019, 11:30 am
Forum: Numerical Methods Forum
Topic: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?
Replies: 327
Views: 12531

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

There's at least one person from the AI community here who's trying to tell you that you're mistaken about the equivalence and explaining clearly why (however difficult it is to pin down what you mean). I am trying to argue with you, on a mathematical basis, concerning this equivalence. But maybe s...
by JohnLeM
April 6th, 2019, 11:01 am
Forum: Numerical Methods Forum
Topic: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?
Replies: 327
Views: 12531

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

Fair enough. Aka indicator/ Kronecker function.
https://en.wikipedia.org/wiki/Dirac_measure

The question is still: "the derivative of a call option is a barrier option". Is that what you mean?
(x-K)^+ ' = {0, x< K, 1, x> K}. Can't I call this a barrier option ?
by JohnLeM
April 6th, 2019, 10:03 am
Forum: Numerical Methods Forum
Topic: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?
Replies: 327
Views: 12531

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

What is a bounded variation function ? it is basically a function which derivative is a measure (even if it is a little bit more complex). Consider a call payoff (x-K)^+. Take its derivative : {0, x < K, 1, x > K}, that is a heavyside function, also called a barrier. Then take a second derivative :...
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