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Cuchulainn
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Why do quants like closed-form solutions?

QuoteOriginally posted by: list1My original question was the meaning of the closed form solution. Others are not important@list is a stickler for definitions; it is not the first time that posters have been caught in his web. Reminds me of my Latin teacher.. You won't succeed:DAnyways, I see closed solution as u = F(parameters) where parameters do _not_ depend on u in any way. In theory we have an explicit solution for u. The function F can be anything.Examples1. du/dt + u = 0 for t > 0 with u(0) = 12. The 2d heat equation in a rectangle using separation of variables3. The GBM SDE with constant coefficientsNon-closed solutions1. Inverse normal CDF2. Nonlinear ODE and SDE3. General Navier-Stokes equation4. your example inverse function $y \, = \, F ( x ) \, = \, \int_0^{\lambda ( x )} f ( t ) \,dt$.But, who knows, you might get lucky finding a closed solution.
Last edited by Cuchulainn on January 16th, 2016, 11:00 pm, edited 1 time in total.
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David Wheeler

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list1
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Why do quants like closed-form solutions?

Thank Cuch I have a similar feeling when I used closed form word. Definitions actually need in order for people who use them understood they are talking about the same issues. Also is it a common notion for English language in mathematics

Cuchulainn
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Why do quants like closed-form solutions?

Of course, having a closed solution may be useless because it cannot be computed.
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David Wheeler

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list1
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Why do quants like closed-form solutions?

QuoteOriginally posted by: CuchulainnOf course, having a closed solution may be useless because it cannot be computed.Cuch, do you have an example that illustrates the statement "having a closed solution may be useless because it cannot be computed", though I asked initially other question whether does the term closed form solution is common notion in math?

Cuchulainn
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Why do quants like closed-form solutions?

QuoteOriginally posted by: list1QuoteOriginally posted by: CuchulainnOf course, having a closed solution may be useless because it cannot be computed.Cuch, do you have an example that illustrates the statement "having a closed solution may be useless because it cannot be computed", though I asked initially other question whether does the term closed form solution is common notion in math?4. your example inverse function $y \, = \, F ( x ) \, = \, \int_0^{\lambda ( x )} f ( t ) \,dt$.does not have a closed solution (IMO) It is nonlinear, which makes life difficult.
Last edited by Cuchulainn on January 16th, 2016, 11:00 pm, edited 1 time in total.
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David Wheeler

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Collector
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Why do quants like closed-form solutions?

I always thought about it the other way around. That it was the closed-form formulas that liked certain people, not always so easy to escape the formulas. Some of them luckily low maintenance .
Last edited by Collector on January 16th, 2016, 11:00 pm, edited 1 time in total.

Cuchulainn
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Why do quants like closed-form solutions?

QuoteAn equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set. For example, an infinite sum would generally not be considered closed-form. However, the choice of what to call closed-form and what not is rather arbitrary since a new "closed-form" function could simply be defined in terms of the infinite sum. [...] In particular, the hypergeometric function (and hence, any closed-form function inheriting its properties) is considered a "special function" and is not expressible in terms of operations which are typically viewed as "elementary." What's more, certain agreed-upon truths like the insolvability of the quintic fail to be true if one extends consideration to a class of functions which includes the hypergeometric function, a result due to Klein (1877).
Last edited by Cuchulainn on January 16th, 2016, 11:00 pm, edited 1 time in total.
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David Wheeler

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list1
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Why do quants like closed-form solutions?

Thanks Cuch. It is quite clear.

Cuchulainn
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Why do quants like closed-form solutions?

So, it seems that a closed form solution can be explicitly written as u = RightHandSide (RHS)where RHS is independent of u. RHS might be easy or difficult, but that is beside the point.Collector has closed for formulae in his book for 2-asset option prices based on exp(x) and the bivariate cumulative normal. The latter has no closed form solution so it must be done numerically, e.g. using Genz, A&S 26.3.3, or by a PDE.
Last edited by Cuchulainn on January 17th, 2016, 11:00 pm, edited 1 time in total.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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list1
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Why do quants like closed-form solutions?

solution represented sounds too general in terms of functions and mathematical operations and seems can be applied for any form of a solution of a problem
Last edited by list1 on January 17th, 2016, 11:00 pm, edited 1 time in total.

Cuchulainn
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Why do quants like closed-form solutions?

QuoteOriginally posted by: list1solution represented sounds too general in terms of functions and mathematical operations and seems can be applied for any form of a solution of a problemIndeed! That's why mathematics is so powerful.This is high-school stuff and very useful QED
Last edited by Cuchulainn on January 17th, 2016, 11:00 pm, edited 1 time in total.
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David Wheeler

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Cuchulainn
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Why do quants like closed-form solutions?

What about finding y(x) that minimizes the length of the curve: Is there a closed solution?
Last edited by Cuchulainn on January 18th, 2016, 11:00 pm, edited 1 time in total.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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list1
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Why do quants like closed-form solutions?

QuoteOriginally posted by: CuchulainnWhat about finding y(x) that minimizes the length of the curve: Is there a closed solution?Quote: An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set. First, our problem does not applicable as far as it starts with minimization of the integral but closed form should originated by equation. On the other hand we should look in which terms the solution of the problem is represented.

Cuchulainn
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Why do quants like closed-form solutions?

QuoteOriginally posted by: list1QuoteOriginally posted by: CuchulainnWhat about finding y(x) that minimizes the length of the curve: Is there a closed solution?Quote: An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set. First, our problem does not applicable as far as it starts with minimization of the integral but closed form should originated by equation. On the other hand we should look in which terms the solution of the problem is represented.How are you getting on?
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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list1
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Why do quants like closed-form solutions?

After be called a racist there is nothing really negative I have not deal with. Actually 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.
Last edited by list1 on February 5th, 2016, 11:00 pm, edited 1 time in total.