Background for Stochastic Calculus?

[Ivan_Milken]

Hey guys, I really want to learn stochastic calculus but I don't have a strong background in math. I know the basics of calculus, but I can't even begin to make sense of a book on stochastic calc. Can someone please tell me step by step which topics I need to master in order to even begin studying stochastic calculus?

Thanks in advance!

 

[mkchan]

You should check out "Brownian motion calculus"

 

[kamikade]

Was asking some people here and answer were mostly "measure theory". I also google and see this guide:

"According to J. Michael Steele, professor of stochastic calculus for the world-renowned Wharton School of Business, the minimum prerequisites for his class are probability theory, multivariate calculus, and linear algebra, the last two of which are senior-level, or graduate-level classes. He goes on to say that success in learning the subject also requires a high level of comfort with real analysis (uniform continuity, Cauchy's convergence criterion, integrability, and calculations in inner product spaces) and measure theory"

At my school, "Advanced calculus" and "Real variables" cover those topic that Professor J. Michael Steele mentioned about. I am not sure if you need to learn everything in "Advanced calculus" and "Real variables" in order to understand Stochastic Calculus throughout. However, I guess that a strong math background will never hurt.

Mathematical Analysis, 2nd edition, by T. M. Apostol is used in Advanced calculus. Topics are from chapter 1-14.
RealAnalysis, 4th edition, by H.L.Royden is used in Real variables. Topics are from page 1-130, 253-281 and 308-424. However, a member here told me it is a little bit overkill for MFE. I still want to take this course if I can though.

 

[Shantanu Kumar]

First level course in Analysis,Probability Theory and Probability & Statistics should be sufficient.

 

[bigbadwolf]

Mathematical Analysis, 2nd edition, by T. M. Apostol is used in Advanced calculus. Topics are from chapter 1-14. Real Analysis, 4th edition, by H.L.Royden is used in Real variables. Topics are from page 1-130, 253-281 and 308-424. However, a member here told me it is a little bit overkill for MFE. I still want to take this course if I can though.

Steele's book is tougher than Shreve's. Apostol and Royden is not overkill for Steele's book. Steele is meant roughly for 2nd year graduate students in mathematics.

 

[Yike Lu]

I fondly recall cursing Apostol's name as a freshman. His Calculus book is much different from other Calculus books, and is the required text for Caltech's "Intro to Calculus" class.

 

[bigbadwolf]

I fondly recall cursing Apostol's name as a freshman. His Calculus book is much different from other Calculus books, and is the required text for Caltech's "Intro to Calculus" class.

I've still got the two volumes (right beside me on the floor). Don't know if they're in print any more. And his book on analysis. And his two books on analytic number theory. Few expositors like him.

 

[marina]

I fondly recall cursing Apostol's name as a freshman. His Calculus book is much different from other Calculus books, and is the required text for Caltech's "Intro to Calculus" class.

"If Calculus were required to be a great investor, I'd have to go back to delivering newspapers." - Warren Buffett

you should have read the book Warren Buffett speaks: wit and wisdom from the world's greatest investor By Warren Buffett, Janet Lowe

 

[euroazn]

I fondly recall cursing Apostol's name as a freshman. His Calculus book is much different from other Calculus books, and is the required text for Caltech's "Intro to Calculus" class.

Don't like proofs?

 

[Yike Lu]

Don't like proofs?

Not as a freshman I didn't.

 

[Ivan_Milken]

Thanks a lot for all the help everyone!!! I've got my work cut out for me but I really think it will prove to be a worthwhile endeavor.

 

[martingaletrader]

• learn real analysis level 101
• learn basics of proofs and what is differentiability and integrability
• master basic probability
• combine your knowledge to learn stoch calc
• you should be be very well versed with integration and differentiation - Advanced calculus level. Believe math for physicists should cover it.
• university level math for engineers should be sufficient; but proofs well you can learn them
• get a decent book on real analysis aimed at undergrads.

 

[Amber Roxanna]

So, Baruch's Pre-MFE course in advanced calculus for financial engineering is not sufficient enough to start studying Stochastic ?

 

[Yike Lu]

So, Baruch's Pre-MFE course in advanced calculus for financial engineering is not sufficient enough to start studying Stochastic ?

1) No it is not, but it covers topics that aren't directly stochastic related.
2) Baruch does have Probability Pre-MFE which should at least help.

As far as sufficiency of the probability course, it depends on what level you wish to understand stochastic. A lot of what everybody else has mentioned is prerequisites to have a deep understanding of the mathematical plumbing underneath Stochastic. However, an engineering knowledge wouldn't require so much real analysis. Nonetheless, an understanding of proofs is essential for higher math in general, so I encourage you to study proofs. A book I recommend for this is "How to Read and Do Proofs" by Daniel Solow. It is essential to work through a good number of the exercises to get conversant in proof-speak.

 

[bigbadwolf]

Nonetheless, an understanding of proofs is essential for higher math in general, so I encourage you to study proofs. A book I recommend for this is "How to Read and Do Proofs" by Daniel Solow. It is essential to work through a good number of the exercises to get conversant in proof-speak.

A first hurdle is trying to understand why proofs are necessary in real analysis -- many theorems seem intuitively obvious and it's not clear why they have to be proved (e.g., intermediate value theorem, mean value theorem). Students have to understand that formal proofs are another way of understanding what might be intuitively obvious, and generally represents a deeper and more interconnected way of understanding. For instance the proof of the intermediate value theorem depends crucially on the assumption of continuity on a bounded interval; take away this assumption and one can construct counter-examples showing the intermediate value property doesn't hold. A second hurdle to clear is understanding that what constitutes a proof depends on the era: what were thought to be proofs two centuries ago no longer satisfy.

Proofs in analysis are not the best place to start because their need does not seem crucial -- often they appear to only be pedantically demonstrating what is already intuitively clear. A better place to start is number theory or group theory or maybe areas like combinatorics and geometry.

One book I like is Hadamard's The Psychology of Invention in the Mathematical Field.