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Replying to Topic: measure theory
Created On Fri Jan 10, 03 02:56 AM by toranaga

toranaga
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Fri Jan 10, 03 02:56 AM

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Could someone kindly explain to me WHAT measure theory is and how/where it's pertinent in QF ?

Additionally, is it of such significance that given a chance, I should take a measure Theory course ?

Thank you,
T

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Toranaga Wa

Aaron
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Fri Jan 10, 03 03:16 AM

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We had a debate about this earlier.

The simple answer is measure theory tells you what sets are legitimate. For example, I can integrate the function f(x)=x^2 over the interval (0,2). But I can't integrate it over only rational numbers in that interval, although I could average it over the rationals. I can't either integrate or average over the irrational numbers in that interval.

There are many different measures. All of the everyday ones give the same answers for simple intervals like (0,2) and simple functions like f(x)=x^2.

Measure theory is only used for proofs in finance. Most of us skip over the technical assumptions. When we write papers, we look up the results in math books or papers, and copy the measure conditions the author used. When we lecture on the paper we say, "and the usual conditions apply." Only rarely does it become important.

It's nice to know a little measure theory, but you get that in a good mathematical statistics course. It stretches your brain to learn real measure theory, and that's always good. But don't expect to get rich applying measure theory.

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Aaron Brown

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Fri Jan 10, 03 03:09 PM

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Let me know if I am wrong, but with good ol' Lebesgue measure, you can integrate any function (that is integrable over the interval in question) over just the rationals. The rationals have measure zero and so the integral is zero.

If your measure is just defined on the rationals, you can certainly integrate on just the rationals. Of course, many measures like this are gonna be "goofy". Although, any discrete measure is defined only on the rationals (or a set like them) and integration becomes summation.

There are a few levels of understanding when it comes to prob and stoch calculus

1. e.g. know Ito's formula and can apply (correctly or not)
2. studied both prob and stoch calc, knew the details once but know what references to use if necessary
3. mathematician or probabilist

Level 2 is good. Most of us don't have the inclination, time or skill to get to level 3. Having good references and friends/coworkers you can call upon is often enough. I get the impression that things are just as Aaron mentions. Most finance profs do not know measure theoretic prob or stochastic calc, but you see a good bit in their papers (some do know it, of course). I think they have a good feel for what is necessary, keep references and walk across campus to the math dept. for confirmation if necessary.

I'd certainly audit or sit in on a course. Depending on your goals, you may regret taking it for a grade.

tbonds
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Fri Jan 10, 03 04:45 PM

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I've been preparing myself for taking a course on Measure Theory and Lebesgue integration. I'm reading H.L. Royden, Bartle, et al, just so it won't be the first time I see the material. It takes a while for the ideas to percolate in the brain and for some level of understanding to arise, and sometimes the notation can get you so lost that you don't understand what the proof was trying to establish in the first place. Case in Point: The proof of Vitali's Covering Theorem.

Aaron, based on your emails on this subject here and elsewhere, it sounds like you don't need an M.S. or PhD to succeed in QF. Just enough of the "right" or "most useful" cherry-picked advanced courses would be sufficient. Am I reading you right?

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tbonds
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Fri Jan 10, 03 04:46 PM

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Let me know if I am wrong, but with good ol' Lebesgue measure, you can integrate any function (that is integrable over the interval in question) over just the rationals. The rationals have measure zero and so the integral is zero.

True.

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If you think you've got math problems, I can assure you that mine are much greater.
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Fri Jan 10, 03 05:19 PM

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toranaga,

I didn't really answer your question.

I expect measure theory was developed to have a more general definition of integration. Ordinary (Reimann) integration is very much like integration with respect to one particular measure, Lebesgue measure, which assigns to an ordinary interval the length of the interval M( [0,4/3) ) = 4/3.

Measures assign a real number to a set, so measure theory is very related to set theory. One of the nice things about math is when dealing with abstract ideas, the results can be applied to all "objects" that fill the assumptions.

If you only have one course to take, I think an applied probability course would be good. To seriously consider a measure course, look at "Real Analysis" by Royden. While useful and nice to understand, there is a long list of topics I'd consider more important to doing work in quantitative finance.

tbonds,

I think you probably do need at least an MS. But, not a Phd, although there is a group of jobs that specifically list Phd as the requirement. Even if you don't need teh structure of an MS, you'll be competing with tons of people who do have advanced degrees. In my opinion, you need grad study unless you are unusually talented. The overall consensus seems to be "you can learn the finance and market conventions on the job, but you need grad school to learn (understand) the math and quantitative tools.

tbonds
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Fri Jan 10, 03 05:36 PM

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The overall consensus seems to be "you can learn the finance and market conventions on the job, but you need grad school to learn (understand) the math and quantitative tools.

Agreed.

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If you think you've got math problems, I can assure you that mine are much greater.
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Fri Jan 10, 03 05:42 PM

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tbonds,

Enjoy Royden. It's a great book which I often still pick up to refresh my memory. Folland will be a helpful reference for HW problems.

JabairuStork
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Fri Jan 10, 03 05:51 PM

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I think that if you have a decent understanding of stochastic calculus and probability, as well as some real analysis background, you can teach yourself a fair amount of measure theory. However, you may find that you don't want to because it's not really necessary to "get the job done" in most work environments. As far as learning on the job, I think the reason most people don't learn math on the job but do learn finance on the job is because the workplace is more conducive to that set-up, not because one is harder than the other.

lyuping
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Fri Jan 10, 03 07:44 PM

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Quote
Originally posted by: Aaron

The simple answer is measure theory tells you what sets are legitimate. For example, I can integrate the function f(x)=x^2 over the interval (0,2). But I can't integrate it over only rational numbers in that interval, although I could average it over the rationals. I can't either integrate or average over the irrational numbers in that interval.

Hmmm, I will borrow this example to explain measure theory to no-math major guys,

tbonds
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Fri Jan 10, 03 07:58 PM

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Enjoy Royden. It's a great book which I often still pick up to refresh my memory. Folland will be a helpful reference for HW problems.

I'm getting used to him.
It took a while to get used to the "non-standard", (or maybe old fashioned?) notation that's used.

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If you think you've got math problems, I can assure you that mine are much greater.
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Fri Jan 10, 03 08:05 PM

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certainly, for a quant fin person, the only measure theory worth learning is the part that helps your understanding in prob and stoch calc. However, that is still a lot of measure theory.

I suppose ifyou are working in a quant group, you could pick up the math. However, how do you get into a quant group before having the math? If you do not have the math and you are not surroudned by people who do, I think it will be very hard to learn the math. My point is that it is easier to pick up a book and read about finance (the non-quant stuff) than it is to do with math.

If you agree (???), then it is harder to pick up the math simply because there are less people who know it.

toranaga
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Fri Jan 10, 03 10:40 PM

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Well, guys, measure theory is just one of the many courses I am considering in prep for a M.Stat. Please see :
http://www.wilmott.com/310/messageview.cfm?catid=8&threadid=4241

I am doing my MBA concurrently and that's why I have a rather limited no of Math courses I can take. I am currently taking Analysis 1 and Probability 1 .

I am planning to take the following courses (as a prep) - please see enclosed list. I am consulting with Professors, but would very much appreciate your comments. Please let me know whether I will be adequately prepared to undertake a M.Stat (at one of the top Canadian schools) after completing these courses. Are there any additional courses I should be looking at ?

Thank you,
T

Analysis II (3 credits)
Connectedness and compactness in the reals. Intermediate value theorem; extreme values for continuous functions. Differential and integral calculus; fundamental theorem of calculus; power series.

Linear Algebra I (3 credits)
Matrices and linear equations; vector spaces; bases, dimension and rank; linear mappings and algebra of linear operators; matrix representation of linear operators; determinants; eigenvalues and eigenvectors; diagonalization.

Real Analysis (3 credits)
Metric spaces; function spaces; compactness, completeness, fixed-point theorems, Ascoli-Arzela theorem, Weierstrass approximation theorem.

Statistics (3 credits)
Point and interval estimation; hypothesis testing; Neyman Pearson Lemma and likelihood ratio tests; introduction to correlation and regression.

Advanced Probability (3 credits)
Central limit theorems and law of large numbers, convergence of random variables, characteristic function, moment generating function, probability generating functions, random walk and reflection principle.

Regression and Analysis of Variance Least-squares estimators and their properties. Analysis of variance. Linear models with general covariance. Multivariate normal and chi-squared distributions; quadratic forms. General linear hypothesis: F-test and t-test. Prediction and confidence intervals. Transformations and residual plot. Balanced designs.

Measure Theory (3 credits)
Lebesque measure and integration on the real line, convergence theorems, absolute continuity, completeness of L2[0,1].

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Toranaga Wa

Patrik
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Sat Jan 11, 03 12:10 AM

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It all depends on what you already know.. All courses listed seems to be standard
and useful engineering mathematics (with the exception of measure theory, that would
be an elective if taken at all by engineers). I come to think of two more areas you might
find useful:
- Numerical methods/numerical analysis
- ODE's and PDE's

These would probably be of less value for a statistics program though..

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/Patrik

toranaga
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Sun Jan 12, 03 01:01 AM

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thanks for the suggestions Patrik.

I've gotten similar advice from other people - ODEs and PDEs are not so relevant. Nevertheless, given the importance of PDEs in Finance (modelling .....BS, etc) I think I will take a course or two in PDEs .......perhaps once I start the M.Stat.

Actually what I am worried about is programming skills .......my last computer course was in undergrad (C++ and Fortran) and I hardly remember anything . How crucial are programming skills in an M.Stat program ? Can I just take one of the "applied" courses where various programming components are integrated into the course structure, or should I take a course offered by the comp science dept ?

P.S : don't know how its done in Europe, but save for linear algebra, NONE of the courses on my list were covered in Mech Eng school !

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Toranaga Wa

Edited: Sun Jan 12, 03 at 01:05 AM by toranaga

mj
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Sun Jan 12, 03 09:45 AM

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Quote
Originally posted by: lyuping
<blockquote>Quote
<hr><i>Originally posted by: <b>Aaron</b></i>

The simple answer is measure theory tells you what sets are legitimate. For example, I can integrate the function f(x)=x^2 over the interval (0,2). But I can't integrate it over only rational numbers in that interval, although I could average it over the rationals. I can't either integrate or average over the irrational numbers in that interval.
<hr></blockquote>

Hmmm, I will borrow this example to explain measure theory to no-math major guys,

actually you can integrate over the irrationals: the rationals are irrelevant as they have zero Lebesgue measure.

MJ

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Patrik
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Sun Jan 12, 03 10:21 AM

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Quote
Originally posted by: toranaga
thanks for the suggestions Patrik.

P.S : don't know how its done in Europe, but save for linear algebra, NONE of the courses on my list were covered in Mech Eng school !

hmm.. then what courses in mathematics did you do? sounds odd to me, here in Sweden all engineers (that I know of) of every program does:
- linear algebra
- calculus/analysis in one and many variables
- basic optimization
- basic probability
- basic statistics

and many programs takes:
- complex analysis
- fourier analysis
- vector analysis
- ODE/PDE
- numerical methods/analysis

Perhaps that's a reflection of "everyone" doing anything technical in Sweden studies in an engineering program. Not many take pure math
and such since it has very little recognition in swedish industry. The title civil engineer is almost the only technical title worth having (from
a job perspective). I think perhaps engineering has come to have a bit different meaning because of this. This and the fact that we don't have
undergrad/graduate levels in the same way, we do 4.5 years and get our masters, no degree before that.

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/Patrik

toranaga
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Sun Jan 12, 03 06:33 PM

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well, prob and stats aren't too relevant to mech eng at the undergrad level (most of everything is deterministic, so no stochastic processes). We take: Cal 1, Cal 2, Cal 3, Vector/Matrices/Linear Algebra, , Advanced Cal, ODE, PDE.

however, the Calculus courses are not the same as Analysis courses - we do NOT prove any theorem.

Any thoughts on the programming skills ?

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Toranaga Wa

Aaron
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Mon Jan 13, 03 02:10 PM

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I stand corrected about integrating over the rationals and irrationals. What should have said is that you can't get reasonable and unambiguous answers for these problems without invoking some measure theory.

A better example is trying to integrate the "ruler" function (named because the graph looks like a ruler). f(x) = 1/k where k is the denominator if x is written as a reduced fraction, that is x = j/k where j and k are integers and k is minimal. For irrational numbers, f(x) = 0 because k is infinite. Depending on your choice of measure, you can get a zero or non-zero answer for this problem.

Real Analysis and Linear Algebra are fundamental courses you need for finance or any quantitative field. Your Analysis II course seems to be a strange hybrid of basic calculus and set theory. I'm puzzled by the description. But if it's a second semester of Real Analysis, it can be useful. Statistics is useful for any quantitative field, although we look at things quite differently in finance. Sometimes a little statistics can be confusing when you see the same concepts used another way in finance. So take the statistics, but keep an open mind. Again, your Advanced Probability course seems misnamed. It appears to be a mathematical statistics course, which is more directly useful in finance than general Statistics.

Regression and ANOVA courses are usually bad, and geared toward the social sciences. I doubt you'll learn much useful there, but you may get an inflated idea of your general intelligence and mathematics ability. A lot of the material could be useful, like linear models with general covariance and multivariate normal math, but you'll probably spend the semester interpreting copious computer printout. Measure theory is the opposite, good for the brain, but probably not much practical use.

My advice is to make sure your Real Analysis and Linear Algebra are really good, and learn a little mathematical statistics. Beyond that, try to take courses from smart people who care about their courses. Material is less important than enthusiasm.

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Aaron Brown

Aaron
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Mon Jan 13, 03 02:20 PM

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To do jobs in quantitative finance, you need to be good at solving math problems. That doesn't depend on any specific material, it depends on how your brain works and how you've been taught. For example, Integral Calculus is usually taught as a problem solving course. Differential Calculus is usually taught as a set of rules to apply. There are reasons for that in the nature of the material, but you could teach either course either way.

Some advanced math courses are taught in problem solving ways: students are assigned proofs or computations. Others spend time going through lots of theorems that have already been proved, students are expected to be able to remember the proofs.\

Someone who is good at solving problems can easily pick up a new branch of mathematics. It's like a good card player learning the rules of a new game. Therefore, I urge people to seek out courses that hone their problem sovling abilities, and teach themselves the specific math material they missed in courses.

To get a job in quantitative finance, it helps to have an advanced degree. Ideally, you have some financial experience, an advanced degree in a quantitative field (ideally finance or math) from a name school and a grandfather who founded an investment bank. But every day people get good jobs with none of the above.

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Aaron Brown

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