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Pardon the dumb question if ever. I notice similar questions all sound like "can I get a second master's?" but mine is "do I need?"

I am nearly finished with my master in mathematical finance. I have finished my thesis and have one course left to take.

Edit: I am about to graduate from a third world university and am planning to apply for a PhD in a first world university.

I am interested in taking up a PhD in a different branch of mathematics called stochastic analysis (stochastic calculus. or whatever it's called.) but am wondering if my background is insufficient. If so, I may take up a second master's.

Given my limited background in stochastic analysis and other information (below), can I already apply for a PhD with stochastic analysis for my dissertation topic, or might I need a second master's first?

Some information:

  1. I am not particularly interested in mathematical finance anymore unless it is in a rigorous context rather than with computers, modeling, statistics, non-mathematical finance, simulations, etc. As of right now, I no longer have any plans to go into industry.

  2. My background in Stochastic Analysis is 2 courses on Stochastic Calculus, 1 prerequisite course for Stochastic Calculus (probability) and 2 halves of classes which apply the stochastic calculus.

  3. I don't know about the basics of one kind of stochastic calculus as we were taught mainly the other kind.

  4. My thesis is about credit risk, having mainly to do with statistics, the part of mathematical finance I don't want to go in to, if I were still interested in mathematical finance. Hence, I feel have no background in mathematical research given that the research conducted was mainly to do with statistics and finance.

  5. I have read a sample PhD proposal and have no clue what to put for methodology. I am not quite sure what kind of research methods pure mathematicians have.

  6. Furthermore, while I technically passed the thesis, I scored the lowest passing mark. Meanwhile, I did score well in all the classes relevant to stochastic analysis.

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    I think the answer to your question may depend on more than just "which world" you are applying to. In Europe, entering PhD students often have very advanced, specialized training. In the US, having a master's degree in a closely related field is better than average training for a PhD (but moreover having a more advanced degree is not too highly correlated with success in the program). I think you would certainly get consideration in a US program. On the other hand, "lowest passing mark" does not inspire too much confidence: if your letters are not strong, it will be hard. – Pete L. Clark Mar 24 '15 at 21:29
  • @PeteL.Clark Thanks. You mean I might get into the US if I were to somehow demonstrate other abilities that compensate for "lowest passing mark" and lack of specialized training? I scored high in the relevant classes. – Jack Bauer Mar 25 '15 at 8:15
  • @PeteL.Clark Regarding which world, I was editing just in case it was brought up as apparently it was important in another question. Another question, do you have any idea if the methodology thing I mentioned may be a problem? – Jack Bauer Mar 27 '15 at 17:24
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    Regarding #5, a dissertation in pure mathematics usually consists of a proof of one or more theorems. The research method is "you prove the theorem". I would suggest that in order to be adequately prepared for a PhD in pure math, one should, at a minimum, be comfortable with mathematical proof in a variety of areas, and have enough experience reading the research literature to know what a mathematics paper looks like. If that's not the case, then probably you do need more preparation. – Nate Eldredge Mar 27 '15 at 19:27
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    A variety of areas throughout mathematics. And that answer of mine that you link was written for an audience completely unfamiliar with mathematical research - someone ready to pursue a PhD should already know what it says. – Nate Eldredge Apr 2 '15 at 13:33
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I would say the answer depends on your background in mathematics, and from the information you give, I can't tell how extensive that is. It isn't so important what the title is on your degree; it matters what you've done and what you know.

To start a US-style PhD in pure math, in which the first couple of years are coursework, you should have the equivalent preparation of a bachelor's degree in mathematics. It should include, for example, about one year each of proof-based real analysis and abstract algebra. You should be very comfortable with reading and writing mathematical proofs.

To start a European-style PhD, you are expected to be ready to begin working on research. You should have completed graduate-level courses in algebra and analysis, and be very familiar with fundamental material in your subfield. For an area like stochastic analysis, for example, I'd say you'd want to know the majority of the material in a textbook like Karatzas and Shreve, and be able to work through the proofs and solve the exercises.

One question is - what was the level of mathematical rigor in your courses up to stochastic calculus? If it wasn't high, that's likely to be a problem.

  • 1 Thanks Nate. That does seem to explain why some of the UK and other European schools I heard of require PhD proposals involving frickin literature reviews while some American schools don't. 2 For the US part: My bachelor's is mathematical finance. We didn't have a thesis because the program assumes most of us take a master's in mathematical finance which then has a thesis (which isn't mathematical). I am very comfortable reading and writing mathematical proofs. I haven't had abstract algebra. :( – Jack Bauer Apr 9 '15 at 16:18
  • 3 For the Eur part: Daaaaamn. Any idea why European schools are so tough? That is one of the books referenced in my Stochastic Calculus I and II. I plan to start reading that along with others ( math.stackexchange.com/q/1204999/198044 ) sometime this week. But is that really enough? Just because I am familiar with the material doesn't mean I can do research well, does it? Or is this kind of post-master's, pre-PhD self-studying the substitute I need for a thesis? – Jack Bauer Apr 9 '15 at 16:21
  • 4 On Rigor: ugh...I had 1 course on Advanced Prob. We used some chapters of Prob w/ Martingales by Williams. StoCal I and II were not very rigorous looking back. Mostly proving results used in finance. We did discuss Our profs didn't discuss convergence of random variables, or go into Radon-Nikodym derivative stuff, but those things were mentioned. They also didn't prove prove Ito's lemma or Girsanov Theorem. As I recall, we proved stuff involving simple processes or functions but didn't prove further than that. Afaik, my program doesn't assume its students will continue to PhD unlike MS Math. – Jack Bauer Apr 9 '15 at 16:25
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    I can't answer all of this right now, and to be honest, I think you really need a long talk with some mentor in the discipline who knows you and your preparation. I can't be your mentor over the Internet. But just in general: European programs are not inherently "tougher", they just assume that you are further along (have a masters in the field). US PhD programs are designed for students to start immediately after a bachelor's degree, so naturally the amount of background knowledge demanded is less. – Nate Eldredge Apr 9 '15 at 16:31
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    The first two years of a US PhD program would get you to roughly the same point at which you would start a European PhD, so US PhD programs take about two years longer overall. It's true that knowing the material in a book like Karatzas and Shreve does not by itself imply you are ready for research - but research in the field will require intimate understanding of those ideas, so if you don't know the material you are very likely not ready. This knowledge is necessary for research but not sufficient. – Nate Eldredge Apr 9 '15 at 16:33
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I'm an engineering graduate student, so things may be different for maths. That being said, I don't think it will be an issue (at least in the US). You should check the admission requirements for the universities you are interested in applying to. I made a similar switch within my major between master's and PhD and nobody cared.

  • Thanks. If I may ask, what was your major in engineering and your major in PhD? (major or thesis topic or dissertation topic, whatever the term is hehehe) – Jack Bauer Mar 25 '15 at 8:16
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    My master's was in themo/fluids and my PhD is in robotics (pretty close to zero overlap between the two). I don't think the admission office would really care (I don't think they would even look at your thesis title). I think all they would care about would be what your major was. Your prospective adviser may care though. – somerandomdude Mar 25 '15 at 17:51
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    "My master's was in themo/fluids and my PhD is in robotics (pretty close to zero overlap between the two)." --> Wicked. "I don't think the admission office would really care (I don't think they would even look at your thesis title). I think all they would care about would be what your major was. Your prospective adviser may care though." --> As I thought. Thanks Hadi. Here's a funny video: youtube.com/watch?v=EkwAzoiMnpw – Jack Bauer Mar 25 '15 at 17:59
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    I'm not sure I agree with this answer as it applies to math. It's not essential that the fields be the same, but it is essential that you have background equivalent to someone with a bachelors/masters in mathematics. And one doesn't tend to encounter rigorous proof-based mathematics in other fields. People with degrees in, say, physics, do sometimes go on to grad school in mathematics, but only if they studied a significant amount of math, more than a typical physics degree would require. – Nate Eldredge Mar 28 '15 at 3:55
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    As Nate mentioned since engineering and math are pretty different this may not apply to you. However, there is barely any overlap between the methods involved in fluid mech and robotics. I pretty much had to start from scratch. – somerandomdude Mar 28 '15 at 19:53
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Just a small clarification on why Europe and US is different (to what mister Pete L. Clark said): The duration of the stay, in Europe it is 3 years and in the US it is 5.

That being said, if you go to Europe, in my honest opinion, you should do at least a master 2 (second year of the master studies) first. They recommend this at the university that I am in and they have a good reason for it (even thought it might sound stupid at first). That way you will meet the people who you might do a PhD with, you will be able to adapt to the environment (if you are not that strong to start off a master 2, go with a master 1 even if it is easier, it gives you time to adapt and adaptation is KEY. Not only that you will study there, you will have to live there for the next few years at least). On the other hand, they have applicants from all over the world. They cannot get a good picture of what mark means what in each country or university.

Conclusion: If it is Europe, look at the tests/exams that Master 1 (contact the director of Master 1 studies if the tests are not available online) students have to pass through, if you think you can solve them with little problem* then go for Master 2. If not, go for Master 1. Good luck!

*At some universities (especially in France) tests are made so that they cannot be fully solved in the given time. That is why you can sometimes solve 2/3 of the test and get 18 out of 20 as a final grade.

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    What is a master 1 and a master 2? – Tobias Kildetoft Mar 28 '15 at 7:41
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    Yes, I am familiar with the European system (as much as there is a uniform one). Those terms, however, are not ones I have come across before, so I doubt if they are standard. – Tobias Kildetoft Mar 28 '15 at 12:28
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    I don't get it. What's a master 1 and master 2? How can someone take only a master 1 if they refer to year levels in a master's degree? – Jack Bauer Apr 2 '15 at 13:18
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    @Jack Please read the explanation that I gave to Tobias regarding the Master 1 and Master 2. I said "only Master 2" because if you feel ready, most European universities will let you start directly from the second year of the master studies. And when I said "go for Master 1" what I meant was for you to start at Master 1 and then do Master 2 hence completing your MSc studies. I hope this helps. – Marko Karbevski Apr 5 '15 at 16:30
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    Pretty much all French universities, each one that you will find here: shanghairanking.com/SubjectMathematics2014.html They are all state schools though. – Marko Karbevski Jul 26 '15 at 22:20

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