How to learn stochastic calculus?

Jakelaker

Active Member
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

Active Member
I'll be learning stochastic processes first but unsure which book to learn from.

a) Introductory to Probability Models - Sheldon Ross
b) Stochastic Processes (2ed) - Sheldon Ross

Book a) has more fundamental concepts but b) has a section dedicated to martingales and seems more advanced. They both looks the same, just one is more advanced and skips (or has a briefer treatment of) more fundamental topics, like random variables.
 

yetanotherquant

Well-Known Member

Jakelaker

Active Member
I highly recommend Stochastic Calculus for Finance II: Continuous-Time Models by Steven Shreve. Ten years ago I managed (after a long break in my mathematical education) to learn stochastic calculus with this book.

As to the measure theory, well, all of my co-students managed to do without but still I highly recommend to have a look at my very readable notes on it.
(you will in either case need it if you come to T-Forward measures in LIBOR Market model).
Could you elaborate on the pros and cons (if there are any) on that book by Steven Shreve? I'm not yet convinced by an anecdote like that. Although that is a great accomplishment, it may not apply to me.

Thanks for the recommendations though, I'll have a look and consider them.
 

Daniel Duffy

C++ author, trainer
I highly recommend Stochastic Calculus for Finance II: Continuous-Time Models by Steven Shreve. Ten years ago I managed (after a long break in my mathematical education) to learn stochastic calculus with this book.

As to the measure theory, well, all of my co-students managed to do without but still I highly recommend to have a look at my very readable notes on it.
(you will in either case need it if you come to T-Forward measures in LIBOR Market model).
Honestly, the chapter 3 looks like a rehash of stuff from other sources (it also contain non-standard notation (symbol for union)?). The Web is full of pdfs like this.
Someone without a solid in pure maths will not understand chapter 3. Just no way. Chapter 4 is just definitions, work in progress?

I have not seen _any_ book that manages to link Measure to Finance in any meaningful way. One exception is the honest shot

"Elementary Stochastic Calculus" Thomas Mikosch.

Shreve and Karatzas is incredibly tough going.

The best book IMO on Measure is by Paul Halmos.
 
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yetanotherquant

Well-Known Member
Could you elaborate on the pros and cons (if there are any) on that book by Steven Shreve?
Only my own experience, sorry.


Honestly, the chapter 3 looks like a rehash of stuff from other sources (it also contain non-standard notation (symbol for union)?).
Yes, I combined (in my opinion) the best approches from different sources.
"|_|" means the union of disjoint sets. If you talk about a (sigma)-additivity, you need to work with disjoint sets.

The Web is full of pdfs like this.
Ach, really? Show my another pdf where the interplay of probabilistic and analytic approaches is shown like this. :)
Or, more precisely, name at least one book or script in which it is shown how to construct a semiring on the paths of Wiener Process.

Someone without a solid in pure maths will not understand chapter 3. Just no way.
Unless one reads chapter 2 first.
And btw, most books on financial math start in chapter-3 like style.

Chapter 4 is just definitions, work in progress?
No. Will probably be resumed as I retire but so far I do more practical things (mostly under P-measure, not Q-measure).


"Elementary Stochastic Calculus" Thomas Mikosch.
Yes, it is not bad.

Shreve and Karatzas is incredibly tough going.
Yes. (there is a great contrast to easy-going Shreve's book I named above.)

The best book IMO on Measure is by Paul Halmos.
For a pure mathematician, maybe. But for a financial engineer it is one of the worst books I've ever seen.
 
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Jakelaker

Active Member
Agreed but that is a different book to Shreve's "Stochastic Calculus for Finance."

Sheldon Ross is a mediocre writer who produces mediocre and dull books.
What aspects of Sheldon Ross' books make them mediocre and dull?

Do you have a recommendation for learning the basics of stochastic processes?
 

Daniel Duffy

C++ author, trainer
No. Will probably be resumed as I retire but so far I do more practical things (mostly under P-measure, not Q-measure).

By the time you retire your mathematical brain will have vaporized. This is the kind of stuff to learn at 18-19. After that it is too late.

Or, more precisely, name at least one book or script in which it is shown how to construct a semiring on the paths of Wiener Process.

I think many people will not lose sleep over this one.
 
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Daniel Duffy

C++ author, trainer
I have never been asked by a quant "Tell me measure theory".

It's not computable/constructive.

One major exception is Radon-Nikodym parameter estimation with MLE as discussed on other forum.
 

yetanotherquant

Well-Known Member
I have never been asked by a quant "Tell me measure theory".
I think those who read this forum regularly are aware that our views on measure theory differ.
In a link on wilmott you provided above Alan says "I think I can safely guess that no MFE graduate understands the Girsanov Theorem". (With an exception of myself) it seems to be true but whether Jakelaker wants to be like this is upto him.

Since a lot of quantitative stuff is about risk-neutral pricing I think every quant should understand the ideas behind the risk-neutral measure, both in economic (market price of risk) and mathematical (Radon-Nikodym, Girsanov) sense. But everyone should decide for him/herself.

It's not computable/constructive.
Not constructive?
How about this? ;)
 

Daniel Duffy

C++ author, trainer
Here's a nice quiz/test: construct the Lebesgue integral of x(1-x) on the interval [0,1] by hand. Take N = 4 steps and calculate upper and lower sums as sums of measurable sets. You might have to solve quadratic equations.

yes? anyone?
 
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Philip_Mitterrand

Well-Known Member
You should be browsing the library picking up diff books and finding the one that fits your level where economy of effort is proportional to knowledge gained.




I'll be learning stochastic processes first but unsure which book to learn from.

a) Introductory to Probability Models - Sheldon Ross
b) Stochastic Processes (2ed) - Sheldon Ross

Book a) has more fundamental concepts but b) has a section dedicated to martingales and seems more advanced. They both looks the same, just one is more advanced and skips (or has a briefer treatment of) more fundamental topics, like random variables.
 

Daniel Duffy

C++ author, trainer
And no one has mentioned Kloeden and Platen's books..

And this one is super cool (I used both it and Rudin I, II, also excellent)

https://www.amazon.com/Problems-Variables-Lebesgue-Integration-Applications/dp/0070602212


from Amazon
One of the most interesting aspects of this book is the development of the Lebesgue Integral in terms of Upper and Lower sums .Too often people do not understand the more abstract definition given in just about all other textbooks.This definition is parallel with that of the Riemann Integral from undergratuate anaylsis.This book is a true gem and a rare find.
 
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Daniel Duffy

C++ author, trainer
Do they use Daniell integral approach? If yes, avoid it! It is elegant but vicious for quants since by this approach the measure is constructed from integral (it is not what we encounter in financial math).
No, I said Lebesgue, not Daniell. AFAIR all these integrals have infinite total variation, making them unsuitable for Wiener increments..


There are zillions of integration methods, each one named after its inventor.

IMO quants use

Riemann
(Riemann Stieltjes?)
Lebesgue
Ito
(Stratonovich)

integrals. It's not a university.

McShane
A Riemann-Type Integral that Includes Lebesgue-Stieltjes, Bochner and Stochastic Integrals
 
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Daniel Duffy

C++ author, trainer
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)
 

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vertigo

Active Member
I have never been asked by a quant "Tell me measure theory".

It's not computable/constructive.

One major exception is Radon-Nikodym parameter estimation with MLE as discussed on other forum.

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. one only needs to look at Hardy's comments about number theory as evidence (constructive proof!) of this stupidity. sometimes even i think like that: "this is not constructive". and then i think of Hardy.

if we use your logic, you would have to present a constructive proof of the real numbers. why? because quants use continuous time models, i.e. [0,1]. how do you know that [0,1] exists? actually you dont. its obvious, right? if you are set on being constructive in mathematics, you will be very miserable. i am not even going to get into constructing the wiener process on [0,1]. guess what though - all proofs of the existence of [0,1] and the wiener process on [0,1] use measure theory (and set theory) at some point. yet we use these objects every day and are often asked about them. do quants ask you to prove the existence of the wiener process? no. do they ask you about properties of the wiener process? yes. usually they ask because the properties can be exploited by a computer. but the properties follow from the definition. you would have to be a downright fool to look at the properties but not the definition, the construction, etc, of the wiener process. and all of that 'deeper' thinking is usually ticked off in measure theory / functional analysis / stochastic analysis courses.

the same holds in mathematical finance. guess what, nobody uses the Black Scholes model anymore - quants use the Black Scholes pricing equation to recover derivative prices given an implied volatility parameter, and also to hedge (functional derivatives of pricing equation). how do we know there is an underlying process that replicates that derivative? unless we impose extreme constraints (which dont hold in reality), we don't. so we are not really being constructive when modelling, either. maybe you can call it pseudo-constructive - because you can perform some MLE estimation and find an 'optimal' value of a parameter, you call this constructive. but the RND arises from some underlying process, and we dont even know if that process replicates the derivative. so why is that estimate useful? It might not be. often it is totally useless. this kind of knowledge can help when completing modelling work as it gives you theoretical perspective.

actually, quants are interested in measure theory, but it is not so obvious. many quant interview questions (for example, coin tossing experiment given some a priori information) are essentially questions on common sense + integrals + filtrations. they aren't written down like that on paper - but knowledge of those topics helps. derivatives that have forward skew/smile, like cliquets, etc, require functional derivatives (measure theory & functional analysis). generally any pay off with convexity will require measure theory at some point, for example using Dominated Convergence theorems to prove that something is submartingale. but this isn't obvious. and its certainly not written down.

and even more generally, the Daniell integral is the best example to understand why measure theory is important. there was a massive rush to present integration following Daniell's approach. guess what the clever mathematicians realised? on a simple level, the Daniell integral is easier to construct than the Lebesgue integral. but any analysis will require proving the same theorems that you do for the Lebesgue integral in measure theory just now for the Daniell integral. And to prove these theorems is a nightmare, because you are no longer working with measures. nor is intuitive. so guess what happened? people accepted measure theory. it definitely has a problem with being constructive, but nothing else is better. it is all that we have. after the Riemann integral of course, which in theory is completely useless but in practice is all that is used.
 
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