How to learn stochastic calculus?

[Daniel Duffy]

after the Riemann integral of course, which in theory is completely useless but in practice is all that is used.

LOL, Archimedes would roll over in his grave were he alive today. He invented all this stuff and the 20th century mathematicians formalized it.

you are missing the purpose of measure theory. some people think as follows: if math topic X is not constructive, it is useless for application, therefore it is pointless to learn it. of course, this is naive and stupid. [...]

I doubt it, but you could be right, During my 9 years in university (3 maths degrees) I never had reason to apply measure theory, with the exception of the Rudin stuff and probability in Banach space. And my lecturer had been a PhD graduate of William Feller at Princeton. Maybe he did not tell us everything.
For me the two least applicable subjects were MT and Category Theory. And awfully boring.

Lebesgue measure is fine; why the smoke screen with fancy Daniell integrals (which no one one uses, correct me if I am wrong. Maybe accept defeat, it's like VHS versus BETAMAX.
The Bourbaki schools loves all that axiomatic mathematics.

Anyways, I'm beyond convincing; it's all those MFE folk who need more help.

To be honest, the discussion is becoming tedious, especially when it becomes ad hominem. Would you say the same to a paying customer?

 

[Daniel Duffy]

that 'deeper' thinking is usually ticked off in measure theory / functional analysis / stochastic analysis courses.

What's deeper thinking?
For Functional Analysis at least (and its applications, e,g. Numerical Analysis, FEM) measure theory is neither necessary nor sufficient, of course depending how you approach the subject.

I did a random Google and found this interesting piece

Measure theory is a must

One claim that I hear often is that one cannot study Functional Analysis or Harmonic Analysis without learning Measure Theory first, or at least concurrently. As someone who has taught functional analysis, operator theory and harmonic analysis a few times, I certainly see the usefulness of knowing some measure theory before hand. It is very convenient for me as a lecturer, especially if I want to use a standard textbook.

However, this claim, although it has some truth in it, is mostly false (it is opposed to experience and to mathematical logic), and is based solely on the personal experience of the claimer (“that’s how I learned it”).

e.g. MT is not necessary in order to understand L2 space. It is also not sufficient.

 

[Daniel Duffy]

What has not been mentioned is solving Stochastic Differential Equations (SDE). In general closed solutions don't exist and they must be solved numerically, for example when computing in Monte Carlo option pricing.

IMO this is an area that is not given much attention in university courses?(?), with the exception of Euler's method

Some SDEs (e.g. GBM, square root CIR) can be exactly simulated.

This is a nice hands-on approach

https://www.amazon.com/Numerical-Solution-Computer-Experiments-Universitext/dp/3540570748

 

[rajanS]

What's your background btw? Are you coming from a business background or engineering background?

Why do you want to understand stochastic calculus?

Should I be learning it from Sheldon Ross' textbook to develop intuition, then move onto a more rigorous approach by learning measure theory then stochastic processes Grimmett's?

I'm not sure what to do. Can someone give me advice on what I should learn in which order?

Thanks.

 

[Jakelaker]

What's your background btw? Are you coming from a business background or engineering background?

Why do you want to understand stochastic calculus?

Maths (pure & stats) and computer science.

After getting obliterated by a stochastic processes class, something clicked and I am really motivated to rel-learn stochastic processes, properly. Once this is achieved, I'll need to learn stochastic calculus so I can operate on and do stuff with stochastic processes.

Secondly, I would like to prepare for future courses relating to the above and a quantitative finance career.

NB: this spontaneous click has occurred to me in the past, with coding (C). Then I worked hard and now am a confident C programmer.

 

[rajanS]

Maths (pure & stats) and computer science.

After getting obliterated by a stochastic processes class, something clicked and I am really motivated to rel-learn stochastic processes, properly. Once this is achieved, I'll need to learn stochastic calculus so I can operate on and do stuff with stochastic processes.

Secondly, I would like to prepare for future courses relating to the above and a quantitative finance career.

NB: this spontaneous click has occurred to me in the past, with coding (C). Then I worked hard and now am a confident C programmer.

Try "Karl sigman " stochastic process notes. He's a professor from Columbia and his notes on that subject are pretty good. It follows Sheldon Ross's book "probability models" which is a beginner stochastic provess book.

To learn stochastic calculus in a pure quant way, you could take an real analysis course for a semester or two that covers measure theory as well.


Hope that helps

 

[Daniel Duffy]

Maths (pure & stats) and computer science.

After getting obliterated by a stochastic processes class, something clicked and I am really motivated to rel-learn stochastic processes, properly. Once this is achieved, I'll need to learn stochastic calculus so I can operate on and do stuff with stochastic processes.

Secondly, I would like to prepare for future courses relating to the above and a quantitative finance career.

Many people have similar questions.
The approach to teaching stochastics could be extended IMO to give it more appeal and more relevance.

1. Don't start with/jump into axioms.
2. Give a 'prologue' course for students who don't have a MSc in maths.
3. Many, many more concrete examples and applications from recent financial literature.
4. Numerical solutions of SDE.
5. Program it up to get numbers in your favourite language.

AFAIK you need to know a bit of Measure but IMO it is a bit exaggerated. As I mentioned, most quants I know have other concerns. Just have a look at the Wilmott site.

 

[kmonks]

Should I be learning it from Sheldon Ross' textbook to develop intuition, then move onto a more rigorous approach by learning measure theory then stochastic processes Grimmett's?

I'm not sure what to do. Can someone give me advice on what I should learn in which order?

Thanks.

I've read a lot of books but nothing beats Steven Shreve. He takes you from a very basic level to a very advanced level.

 

[Daniel Duffy]

Here is a book by Sean Dineen whom I knew back in Dublin. He is a very good mathematician and it might be worth a look at

Amazon.com: Probability Theory in Finance: A Mathematical Guide to the Black-scholes Formula (Graduate Studies in Mathematics): 9780821894903: Seán Dineen: Books

Amazon.com: Probability Theory in Finance: A Mathematical Guide to the Black-scholes Formula (Graduate Studies in Mathematics): 9780821894903: Seán Dineen: Books

www.amazon.com

 

[binomial-torrent]

i find reading shreve alone is very boring and ineffective. you will learn more and quickly if you can find lecture slides from cmu or columbia.

link?

 

[Daniel Duffy]

here is a discussion from a while back

How much measure theory do you need for admission to MFE? - wilmott.com

forum.wilmott.com

 

[ibyte]

The book of Oksendal "Stochastic Differential Equations" is a good intro. The standard models in finance show up in examples and exercises although there is no discussion of finance.

 

[PepeQuant]

The book of Oksendal "Stochastic Differential Equations" is a good intro. The standard models in finance show up in examples and exercises although there is no discussion of finance.

LMAO. your intro will scare them all away bro... perhaps it is an intro for some folks but alien lang to others

 

[Daniel Duffy]

LMAO. your intro will scare them all away bro... perhaps it is an intro for some folks but alien lang to others

I agree. It is NOT an introduction!
(I did about 4 undergrad courses on this + applications and it is still a challege).

Plan B: KLoeden and Platen,

 

[kmonks]

Here is a book by Sean Dineen whom I knew back in Dublin. He is a very good mathematician and it might be worth a look at

Amazon.com: Probability Theory in Finance: A Mathematical Guide to the Black-scholes Formula (Graduate Studies in Mathematics): 9780821894903: Seán Dineen: Books

Amazon.com: Probability Theory in Finance: A Mathematical Guide to the Black-scholes Formula (Graduate Studies in Mathematics): 9780821894903: Seán Dineen: Books

www.amazon.com

I borrowed this book from a library and I have to agree, it's a very good introduction to the subject!

 

[Daniel Duffy]

I borrowed this book from a library and I have to agree, it's a very good introduction to the subject!
In fact, I would even go on to say if you don't want a very comprehensive assessment of stochastic calculus, but just enough to work your way to the Black-Scholes model, this book is brilliant!

I just bought my own copy and it arrived yesterday. It is much better and lucid than a lot I have seen down the years.
And Sean Dineen is a top pure mathematician.

Some background on SD is here

https://www.maths.tcd.ie/pub/ims/bull64/M6403.pdf

 

[Quasar Chunawala]

Here are some slides that I created on Measure and Lebesgue integration + numerical example + useful theorems.

Yesterday, I just tried to integrate y = 4 * x * (1-x) on the interval (0,1) (exact value 2/3) using the slides, with pencil and paper and a 90s TI calculator.

We get an interval solution that converges (is very instructive as new insights emerge)


n= 2 (.3522, .8530)
n = 3 (.4827, .7982)
n = 4 (.5183, .7694)

It was quite instructive to work this out.