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Topic Title: measure theory
Created On Fri Jan 10, 03 02:56 AM

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toranaga
Senior Member

Posts: 259
Joined: Jul 2002

Fri Jan 10, 03 02:56 AM

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

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Aaron
Senior Member

Posts: 6371
Joined: Jul 2001

Fri Jan 10, 03 03:16 AM

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

Posts: 194
Joined: Feb 2002

Fri Jan 10, 03 03:09 PM

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.

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tbonds
Senior Member

Posts: 244
Joined: Jul 2002

Fri Jan 10, 03 04:45 PM

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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If you think you've got math problems, I can assure you that mine are much greater.
-Einstein

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tbonds
Senior Member

Posts: 244
Joined: Jul 2002

Fri Jan 10, 03 04:46 PM

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.
-Einstein

Posts: 194
Joined: Feb 2002

Fri Jan 10, 03 05:19 PM

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.

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tbonds
Senior Member

Posts: 244
Joined: Jul 2002

Fri Jan 10, 03 05:36 PM

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.
-Einstein

Posts: 194
Joined: Feb 2002

Fri Jan 10, 03 05:42 PM

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.

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JabairuStork
Senior Member

Posts: 948
Joined: Feb 2002

Fri Jan 10, 03 05:51 PM

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.

Posts: 26
Joined: Mar 2002

Fri Jan 10, 03 07:44 PM

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,

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tbonds
Senior Member

Posts: 244
Joined: Jul 2002

Fri Jan 10, 03 07:58 PM

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.
-Einstein

Posts: 194
Joined: Feb 2002

Fri Jan 10, 03 08:05 PM

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.

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toranaga
Senior Member

Posts: 259
Joined: Jul 2002

Fri Jan 10, 03 10:40 PM

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

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Patrik
Senior Member

Posts: 323
Joined: Apr 2002

Sat Jan 11, 03 12:10 AM

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..

-------------------------
/Patrik

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toranaga
Senior Member

Posts: 259
Joined: Jul 2002

Sun Jan 12, 03 01:01 AM

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

Posts: 3006
Joined: Dec 2001

Sun Jan 12, 03 09:45 AM

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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Quant Job Interview Questions and Answers now available on lulu.com and amazon.com

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Patrik
Senior Member

Posts: 323
Joined: Apr 2002

Sun Jan 12, 03 10:21 AM

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 !