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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.
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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.
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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!
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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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.
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David Wheeler

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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]
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David Wheeler

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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 ?"

?
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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?

Maybe only those quants who can't program.
The quote was from Jacques Hadamard, e.g. is a PDE 'any good' without trying to find a solution or taking a specific example?
It's a bit like Linnaeus taxonomy stuff or concept hierarchies (superordinate, basic, subordinate). OOP calls it iiheritance.

Example published article double barrier for cash or nothing using SDE A-Z (a lot of hours work, just look eqs. 27, 28!) Reinventing the wheel??
Hadamard would say it's just a PDE with a discontinuous payoff..Instead of always being quantitative ('"chechez closed solution") first examine qualitative properties.

https://www.researchgate.net/profile/Ja ... ion_detail

BTW PDE solver gives same results w/o any modification to the code and no extra maths  needed.

// Series solutions are a-posterior accurate (like ANNs) and they may not always converge., With FDM, choose step length h, dt a-priori.
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Cuchulainn
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### Re: Why do quants like closed-form solutions?

A final remark on the relative merits of producing analytical solutions versus defining a finite difference framework that subsumes and supports a range of pdes sharing similar structure and functionality:  in the former case each new specific option type will need its own specific formula (at great expense in human resource terms and in terms of the origination and testing of analytical option pricing formulae). Reusability of the algorithm and of code is not guaranteed and the algorithm may place restrictions on the parameters if it is to converge. Furthermore, the accuracy is fixed (and cannot be measure a priori) and it is usually not possible to customise the algorithm to suit difference performance needs. For example, it is difficult to customise the algorithm that produces six-digit accuracy to produce results to two-digit accuracy.  In the latter case (using finite differences) we only need to set up a flexible software framework once and then it can be instantiated for specific types, for example spread options. A priori accuracy is determined by the mesh sizes. This usually involves nothing more than using a different payoff function with no modification of the rest of the code being needed. This is the stage in which we can apply software design patterns to achieve this desired level of flexibility.  Finally, the framework can even be used for problems for which an analytical solution is not forthcoming.
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David Wheeler

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

What's the fascination when some quants want to transform every PDE under the sun to the heat equation?
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David Wheeler

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