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Importance of math rigorousness and other questions

Hello everyone,

I started this thread due to some confusion in my part, i'm somewhat confused on how rigorous and extensive one's math background should be. I understand that the minimum requirements such as: Calculus I, II, III, Linear Algebra, probability, statistics and ODEs but to what extent are they supposed to be covered? sometimes I read mentions about real analysis and measure theory and my question is why? how exactly would someone stand to benefit from these courses?

Another thing I don't understand from employers is the preference for PhDs, I can't quite comprehend how they can bring value or are raise their advantage. They praise the PhDs ability to do research but at the same time those who get industry jobs (irrelevant to their research) could might as well be failed academics, how can you praise their research ability when they couldn't do proper research in their own domain which they're prepared for. Eventually, how can they even depend on their coding abilities if they don't come from a computational heavy PhD that relies on C++ or Python.

What does a pure math or mathematical physics PhD bring to the table and what exactly can they perceive and others don't?
 
No MFE program “requires” prior coursework in real analysis or measure theory, to my knowledge Columbia’s MAFN is the only top program that even mentions measure theory on their website. In many cases, learning stochastic calculus is much easier if you’re comfortable with analysis — this is of course not to say one needs to have such comfort to learn sto cal, but in my experience it makes the process far more enriching. If you know how to construct and have worked with Riemann-Stieltjes integrals (standard in a Real Analysis I course), then Itô integrals will seem relatively tame to you.

Let me say this first: PhDs in finance are not failed academics, far from it. Check this out: Meet a Quantitative Research Intern | SIG. PhDs bring value because they are at the cutting edge of their domain — while MFEs do learn advanced math and stats, it is only the tip of the iceberg and PhD students focused exclusively on such fields will be far more familiar with current research and their knowledge will be specialized. What if they’re lacking a programming background and want to break into quantitative finance? They do as they surely have had to many times over during their PhD, they self-teach themselves! While I am only a masters student in pure math, this was very much the situation I was in. The introductory C++ course hosted here on QN is fantastic, and is perfectly suited to be the driving force behind learning C++ in a self-guided way.
 

Daniel Duffy

C++ author, trainer
Calculus is about learning (useful) tricks. It is has its limits, Real analysis is learning to think like a mathematician. And a lot of students are missing a decent grounding in numerical analysis.

A good PhD can take a completely new problem and solve it from scratch.

Just having a PhD in maths does not imply being good at C++, You have to learn the discipline of programming, which they don't do in the ivory towers. Many academics don't like programming.

If Euler were alive today he would do C++.
 
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Daniel Duffy

C++ author, trainer
If you know how to construct and have worked with Riemann-Stieltjes integrals (standard in a Real Analysis I course), then Itô integrals will seem relatively tame to you.

Nope. It won't help with Girsansov, Radon-Nikodym, Feynman-Kac,...
Lebesgue is better.

I think Dineen is good book. He is a top pure mathematician
 
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Riemann-Stieltjes motivates why nonzero quadratic variation is a necessary hurdle to overcome when constructing the Itô integral. This text — Introduction to Stochastic Integration | Hui-Hsiung Kuo | Springer — makes the importance of R-S integration clear from Ch. 1. After all, nonzero quadratic variation is what makes Itô integrals different. Just saying “Nope. Lebesgue is better.” fails to recognize the motivation R-S integration provides for Itô integrals. I am not claiming such a method of integration is good or bad, simply that it well motivates Itô’s theory for students with only an elementary analysis background.
 
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I should have said, “then *the construction of* Itô integrals will seem relatively tame to you”. Respectfully, I think you are missing my point. I did not say R-S integration is what drives the theory of stochastic calculus, simply that having experience with it makes understanding Itô integrals easier. That’s all.
 
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I went into MFE w/ second BS in Mathematics, which I somehow managed to earn without real analysis. Which was a mistake on my part. Very often, I needed a lot of help from friends to even understand what the proof-based questions were asking. I have the hang of it a bit more, and it's fun, but still a gap.

Also, @Daniel Duffy - thank you for this gem: "Calculus is about learning (useful) tricks. It is has its limits. . ."
 
Calculus is about learning (useful) tricks. It is has its limits, Real analysis is learning to think like a mathematician. And a lot of students are missing a decent grounding in numerical analysis.

A good PhD can take a completely new problem and solve it from scratch.

Just having a PhD in maths does not imply being good at C++, You have to learn the discipline of programming, which they don't do in the ivory towers. Many academics don't like programming.

If Euler were alive today he would do C++.

If Euler were alive today he would do C++.
That is a bold statment but i can't agree more :)
 

Daniel Duffy

C++ author, trainer
I went into MFE w/ second BS in Mathematics, which I somehow managed to earn without real analysis. Which was a mistake on my part. Very often, I needed a lot of help from friends to even understand what the proof-based questions were asking. I have the hang of it a bit more, and it's fun, but still a gap.

Also, @Daniel Duffy - thank you for this gem: "Calculus is about learning (useful) tricks. It is has its limits. . ."
I would say that real analysis is core, everything else felt like an "add-in", the exception being Functional Analysis.
 

Daniel Duffy

C++ author, trainer
Riemann-Stieltjes motivates why nonzero quadratic variation is a necessary hurdle to overcome when constructing the Itô integral. This text — Introduction to Stochastic Integration | Hui-Hsiung Kuo | Springer — makes the importance of R-S integration clear from Ch. 1. After all, nonzero quadratic variation is what makes Itô integrals different. Just saying “Nope. Lebesgue is better.” fails to recognize the motivation R-S integration provides for Itô integrals. I am not claiming such a method of integration is good or bad, simply that it well motivates Itô’s theory for students with only an elementary analysis background.
Where does the R-S break down so to speak?
 
You can find more in the Riemann-Stieltjes section here: P-variation - Wikipedia. This rules out the possibility of defining an integral with respect to Brownian motion as a pathwise Riemann-Stieltjes one (at least, when the integrand is not of finite p-variation with 2>p>=1), which is the naive way a student may guess one could define such an integral when first studying stochastic calculus.

For the record, I agree Lebesgue is better given its ability to handle pathological functions such as the one crafted by Dirichlet, however, Riemann-Stieltjes integration certainly sheds light on why there are inherent difficulties with constructing stochastic integrals.
 
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Calculus is about learning (useful) tricks. It is has its limits, Real analysis is learning to think like a mathematician. And a lot of students are missing a decent grounding in numerical analysis.

A good PhD can take a completely new problem and solve it from scratch.
This is a good summarized answer. If you have foundation in mathematical rigor, you can breakdown complex problems into simpler pieces.

A lot of basic level math courses like Calculus or Linear Algebra gloss over the proofs part, and like Daniel said, only teach you "tricks" (or tools, rather) to solve problems. You can be taught to be an expert in doing complex first-order or second-order differentiations without knowing what it really means, and how to use it to model a real world problem, like say a predator-prey population model.
 

Daniel Duffy

C++ author, trainer
I went into MFE w/ second BS in Mathematics, which I somehow managed to earn without real analysis. Which was a mistake on my part. Very often, I needed a lot of help from friends to even understand what the proof-based questions were asking. I have the hang of it a bit more, and it's fun, but still a gap.

Also, @Daniel Duffy - thank you for this gem: "Calculus is about learning (useful) tricks. It is has its limits. . ."
In this link, parts B,C and D are wonderful foundation for much of the stuff to learn for many studies. It would be a great mini course and I can cook up good examples in C++ and Python.

 
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Daniel Duffy

C++ author, trainer
This is a good summarized answer. If you have foundation in mathematical rigor, you can breakdown complex problems into simpler pieces.

A lot of basic level math courses like Calculus or Linear Algebra gloss over the proofs part, and like Daniel said, only teach you "tricks" (or tools, rather) to solve problems. You can be taught to be an expert in doing complex first-order or second-order differentiations without knowing what it really means, and how to use it to model a real world problem, like say a predator-prey population model.
Not only is analysis 'deeper' than Calculus, it helps to teach how to start thinking like a mathematician. See Polya for a brilliant exposition



Mathematics is not a deductive science -- that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.
Paul Halmos
 
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