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Cuchulainn
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### Why do quants like closed-form solutions?

QuoteOriginally posted by: list1Actually for about a month I look at SDEs from math point of view. I found a simple proof of the well known result and start to think about its generalization. I bumped with a technical difficulty which might be known or not as I am not sure about. It relates may be to closed form solution. It is well known that solution of the linear sde admits closed form representation. Whether does this result is known for two dimensional case? Actually I need the result that states positiveness of the solution if the initial value is positive. I need it for pure mathematics. On the other hand it can be used in finance too as a model in which correlated indexes or currencies are modeled by multidimensional system of SDEs. I will appreciate to a suggested hint.I find the term 'pure mathematics' a bit artificial. Quote states positiveness of the solution if the initial value is positiveThis is a qualitative property of the solution that you should be able to prove without actually having a closed form. It is called the maximum principle in PDE.//BTW the bespoke calculus of variations example and 'solved' by the Euler equation was discovered e as follows:QuoteIn solving optimisation problems in function spaces, Euler made extensive use of this methodof finite differences'. By replacing smooth curves by polygonal lines, he reduced the problem offinding extrema of a function to the problem of finding extrema of a function of n variables, andthen he obtained exact solutions by passing to the limit as n ! 1. In this sense, functions canbe regarded as functions of infinitely many variables' (that is, the infinitely many values of x(t)at different points), and the calculus of variations can be regarded as the corresponding analog ofdifferential calculus of functions of n real variables.
Last edited by Cuchulainn on February 8th, 2016, 11:00 pm, edited 1 time in total.

list1
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### Why do quants like closed-form solutions?

1. I find the term 'pure mathematics' a bit artificial. // Pure math is when one is proving a theorem and do not think about its applications and focus only to express proof shortly and in most comprehensive form.2. This is a qualitative property of the solution that you should be able to prove without actually having a closed form. It is called the maximum principle in PD// It will be good. Also I have not worked much with matrices and actually could not understand quite simple questions. For example if A is a 2$\times$2 matrix. Is it possible to present explicitly elements of of the matrix $e^A$? It might be written somewhere and not a difficult question.3. Its often happen in math that similar problem is simple and well resolved while what one is needed is a problem having not obvious solution.
Last edited by list1 on February 8th, 2016, 11:00 pm, edited 1 time in total.

Cuchulainn
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### Why do quants like closed-form solutions?

QuoteOriginally posted by: list1For example if A is a 2$\times$2 matrix. Is it possible to present explicitly elements of of the matrix $e^A$? It might be written somewhere and not a difficult question.For exp(Matrix) this might be a good startAlso, textbooks on ODEs will usually have a chapter or two on this subject. Best to start with simple examples. It's a real deep theory. It underlies much of PDE/FDM. Depending on the structure of A, it might be possible to find an exact representation for exp(A) etc. Maybe the matrix is diagonalizable, then it becomes easy.
Last edited by Cuchulainn on February 9th, 2016, 11:00 pm, edited 1 time in total.

list1
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### Why do quants like closed-form solutions?

Thanks Cuch for directions.

Cuchulainn
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### Why do quants like closed-form solutions?

You're welcome!

Cuchulainn
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### Why do quants like closed-form solutions?

QuoteIn 1918 Einstein published the paper ÜBER GRAVITATIONSWELLEN [1] in which, for the first time, the effect ofgravitational waves was calculated, resulting in his famous quadrupole formula (QF). Einstein was forced to thispublication due to a serious error in his 1916 paper [2], where he had developed the linear approximation (weakfield)scheme to solve the field equations of general relativity (GR). In analogy to electrodynamics, whereaccelerated charges emit electromagnetic waves, the linearized theory creates gravitational waves, popagatingwith the speed of light in the (background) Minkowski space-time. A major difference: Instead of a dipolemoment, now a quadrupole moment is needed. Thus sources of gravitational waves are objects like a rotatingdumbbell, e. g. realized by a binary star system.
Last edited by Cuchulainn on February 11th, 2016, 11:00 pm, edited 1 time in total.

Cuchulainn
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### Re: Why do quants like closed-form solutions?

There is a third way between searching for an explicit solution (a big favourite!) and a numerical solution

In mathematics, the qualitative theory of differential equations studies the behavior of differential equations by means other than finding their solutions. It originated from the works of Henri Poincaré and Aleksandr Lyapunov. There are relatively few differential equations that can be solved explicitly, but using tools from analysis and topology, one can "solve" them in the qualitative sense, obtaining information about their properties.[1]

Cuchulainn
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### Re: Why do quants like closed-form solutions?

"Can the existence of a mathematical entity be proved without defining it ?"

?

katastrofa
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### Re: Why do quants like closed-form solutions?

Maybe only those quants who can't program.

Cuchulainn
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### Re: Why do quants like closed-form solutions?

Only closed form solutions lend lucid insight.
Lucid and incorrect go hand-in-hand in many cases! You need to quantify the assumptions in your formulas because closed solutions break down when parameters are outside some range. But only heuristics (aka a-posteriori hand-waving) tell us if they are good or not.

We have a large number of mathematically and numerically correct PDE/FDM schemes for a range equity and ir/fixed income (CIR, HW1, HW2) problems (CN,ADE, ADI, ADICS, MOL, Marchuk, Yanenko, Strang, Saulyev, DD exponential fitting.)

Concrete example: one 2d test is Kirk's (1995) approximation as in Haug (2007), pages 213-214. It is good in the main but deep OTM, ITM less so. But all the fdm schemes give the same answers as each other.

We are not use traditional FDM, FFT or MC. Too much hassle.

Similar excellent results HW1, HW2 incorporating instantaneous forward rate.

Next step: a few of to apply HW2 to CMS spread range accruals.

Hypothesis: the best approximation to a closed solution is to apply [3,7] robust FD schemes to the PDE, letting dt -> 0, h -> 0 and checking results are the same. They all can't be wrong because each one is a both a generalist and a specialist.

Cuchulainn
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### Re: Why do quants like closed-form solutions?

Only closed form solutions lend lucid insight.
Not true, in short.

bearish
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### Re: Why do quants like closed-form solutions?

I think there is a flip side to this question, which is why do some people abhor closed form solutions? I seem to recall some barrier option problem we discussed a few years ago and I give you a nice simple closed form solution, and you kept on yammering about needing a non-trivial PDE to solve numerically instead. What’s up with that?

Cuchulainn
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### Re: Why do quants like closed-form solutions?

I think there is a flip side to this question, which is why do some people abhor closed form solutions? I seem to recall some barrier option problem we discussed a few years ago and I give you a nice simple closed form solution, and you kept on yammering about needing a non-trivial PDE to solve numerically instead. What’s up with that?
It is an honour to have made an impression, in between your Trump meanderings

It was probably you had a closed form approximations like this bespoke one

Concrete example: one 2d test is Kirk's (1995) approximation as in Haug (2007), pages 213-214. It is good in the main but deep OTM, ITM less so. But all the fdm schemes give the same answers as each other.

Cuchulainn
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### Re: Why do quants like closed-form solutions?

Closed-form solutions only work in the simplest case and will break down for extreme parameter values, for example etc. etc. Sometimes you can get lucky with these serendipitous discoveries.

In the Kirk case, 5 fd schemes give the same values while Kirk's formula is the outlier.

It is better to solve one problem five different ways, than to solve five problems one way.
- George Polya