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Stochastic Calculus

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Which book would you recommend for a stochastic calculus? I read in this forum that many finds Steven Shreve's Stochastic Calculus for Finance difficult and give up from the scratch. Any ideas about this book? Would you suggest this book over others? Note: difficulty is not a concern, the only important thing is to contain most needed material in this field. Thanks
 
I assume you want to learn stochastic on your own? Shreve is not good for self-study. If you look at the exercises at the end of the chapters they are mostly about the proofs, you need exercises that ask you to, e.g., find the option prices. Also, Shreve skips on some details. I cannot recommend anything else, Shreve is what we use at my program, but professor came up with all the hw problems.
 
I have already learned the elementary stochastic calculus for finance and need something higher.

I assume you want to learn stochastic on your own?

Exactly. We have no such subject at the university at this moment so I want to learn it on my own.
 
I have already learned the elementary stochastic calculus for finance and need something higher.

then I don't see why Shreve would be difficult.

Also, try Bjork's "continuous time finance" book. I like it better than Shreve.
 
I assume you want to learn stochastic on your own? Shreve is not good for self-study. If you look at the exercises at the end of the chapters they are mostly about the proofs, you need exercises that ask you to, e.g., find the option prices. Also, Shreve skips on some details. I cannot recommend anything else, Shreve is what we use at my program, but professor came up with all the hw problems.

Actually that's what attracts me more than finding solution for a particular problem.

then I don't see why Shreve would be difficult.

Also, try Bjork's "continuous time finance" book. I like it better than Shreve.

I like Shreve since I used once for my diploma theme preparation. Nor I think it is very difficult. I have to provide some elementary stochastic knowledge which I already have (mainly) and then move to it. Thank you all
 
Which book would you recommend for a beginer ? some links ? Thanks

Hi. I would recommend these ones:

http://www.amazon.com/Stochastic-Calculus-Finance-Binomial-Springer/dp/0387249680
http://www.amazon.com/Stochastic-Calculus-Finance-II-Continuous-Time/dp/0387401016

I have also started studding these books and looks impressive. There are some opinions against those say that there are few examples and many analytical solutions in end of chapter exercises and inside the chapters for explanation but I think that it is perfect to understand the concepts. This is my opinion: every chapter must prove the concepts presented in analytical way rather than with examples because examples hide the flexibility of dealing with methods. Nor I agree with the opinion that these books are a bit difficult for beginners. Once you have completed the basic courses of calculus and more of statistics, you will not find it difficult at all.
 
This is my opinion: every chapter must prove the concepts presented in analytical way rather than with examples because examples hide the flexibility of dealing with methods. Nor I agree with the opinion that these books are a bit difficult for beginners. Once you have completed the basic courses of calculus and more of statistics, you will not find it difficult at all.

This is your opinion. Some people like examples more than theory beause they want to see the applications. The theory might be really boring. Also, what you consider basic calculus might not be exactly the same as what people in US consider basic calculus.
 
Nor I agree with the opinion that these books are a bit difficult for beginners. Once you have completed the basic courses of calculus and more of statistics, you will not find it difficult at all.

You don't know what you're saying.
 
You don't know what you're saying.

I have also been told many times it's hard for beginners. I myself am a beginner in stochastic calculus. I am learning it now and want to say that I feel quite comfortable with these books. So I know what my experience is. Give me one reason why the above courses covered cannot be enough for starting in mathematical sense. I purchased both books and the third one also "Stochastic calculus and finance" and they can be learned on the basis of calculus and statistics. If you don't agree then tell me if you have read any of them? If you mean there are contents from linear algebra, I would include it into calculus which is enough. It is very preferable to have some financial knowledge and understand investments theories and derivatives which I have covered and find it even more comfortable to learn these books on those grounds but I meant from mathematical point of view since I was discussing the criticism about the ways concepts are explained there - i.e. analytical way or through the examples. So I still suggest this book for beginners (as I am a beginner in this subject) having knowledge of statistics and calculus(including linear algebra).
 
This is your opinion. Some people like examples more than theory beause they want to see the applications. The theory might be really boring. Also, what you consider basic calculus might not be exactly the same as what people in US consider basic calculus.

Alan. I agree that many people like examples but for introductory purpose it is better (in my opinion of course) to give analytical derivation of formulas rather than state already proved formula and give examples straight away. OK this is my opinion which can be a subject of debate but what is not a subject of debate and argument is that examples are limited in capability of presenting the concept with full insight. On my example, I have read a fixed income mathematics book by Fabozzi which I didn't like at all since the examples hide the flexibility of models. When you are presented a model and are told that it is always so, you don't have enough satisfaction since if you wanted to change one variable then you don't know the effects of it on the entire model (this book example again). Ok, let it be my opinion but for me examples are terrible introducers of concepts.
 
For a first look on stochastic processes i would recomend "Basic stochastic processes" of Brezniak and Zastawniak. Its simple to understand and covers a lot of theory with solved exercises.

http://www.amazon.com/Basic-Stochastic-Processes-Undergraduate-Mathematics/dp/3540761756/

If you are interested in financial methods based on stochastic calculus, i think that : Arbitrage theory in continuous time" by Tomas Bjork, is probably one of the best introductory books.

http://www.amazon.com/Arbitrage-Theory-Continuous-Oxford-Finance/dp/019957474X/

For a more advanced study, "Mathematical methods for financial markets" by Jeanblanc, Yor and Chesney, is the best available now.

http://www.amazon.com/Mathematical-Methods-Financial-Markets-Springer/dp/1852333766/
 
If you don't agree then tell me if you have read any of them? If you mean there are contents from linear algebra, I would include it into calculus which is enough. It is very preferable to have some financial knowledge and understand investments theories and derivatives which I have covered and find it even more comfortable to learn these books on those grounds but I meant from mathematical point of view since I was discussing the criticism about the ways concepts are explained there - i.e. analytical way or through the examples. So I still suggest this book for beginners (as I am a beginner in this subject) having knowledge of statistics and calculus(including linear algebra).

Your statement that "it is not difficult for someone who has taken calculus and statistics" is wrong. You are playing with words. And what you are saying can give the wrong impression to someone who may be reading what you write. Shreve is demanding -- particularly volume 2. Shreve has simplified the presentation to the extent possible -- the book is not comparable in difficulty or abstraction to the two volumes of Karatzas and Shreve. But it is still involves hard work, and even though he explains terms like Borel measures and sigma-algebras, a students would still need to bring considerable mathematical maturity to his study. Something going well beyond calculus and statistics.

Also, Alain is right that examples are key. Generally when starting a new subject area in math, we begin with clear examples. This holds in algebra, in mechanics, in number theory, in differential geometry -- everywhere. The theory comes later, serves to glue together the disparate examples, and serves to explain phenomena on a more abstract level. Thus for example I can give the helix as an example of a curve with both curvature and torsion. Only later will I give a general proof that every space curve can be completely characterised by curvature and torsion. Only the worst teachers begin with an abstract and unmotivated exposition.

For these reasons you do not know what you are saying.
 
Your statement that "it is not difficult for someone who has taken calculus and statistics" is wrong. You are playing with words. And what you are saying can give the wrong impression to someone who may be reading what you write. Shreve is demanding -- particularly volume 2. Shreve has simplified the presentation to the extent possible -- the book is not comparable in difficulty or abstraction to the two volumes of Karatzas and Shreve. But it is still involves hard work, and even though he explains terms like Borel measures and sigma-algebras, a students would still need to bring considerable mathematical maturity to his study. Something going well beyond calculus and statistics.

Also, Alain is right that examples are key. Generally when starting a new subject area in math, we begin with clear examples. This holds in algebra, in mechanics, in number theory, in differential geometry -- everywhere. The theory comes later, serves to glue together the disparate examples, and serves to explain phenomena on a more abstract level. Thus for example I can give the helix as an example of a curve with both curvature and torsion. Only later will I give a general proof that every space curve can be completely characterised by curvature and torsion. Only the worst teachers begin with an abstract and unmotivated exposition.

For these reasons you do not know what you are saying.

As for examples and theory, I still stay on my opinion since I can also provide sequence in many mathematical concepts where theory is being backed by examples later but this is not as important to me. I DO NOT neglect the usefulness and necessity of examples but what I don't like is people saying examples are enough - this is what is actually meant by not liking analytical introduction of concepts and by preferring to have stated examples since I have come across many limitations in examples ALONE. That's why I hate examples as tools of introduction. I think you got this in another way. You mean: it is better to see the problem first and then find the theoretical solution which itself implies bringing examples first and then presenting theory. I didn't mean that. Have you seen the book I mentioned?! (Fabozzi fixed income mathematics). It is full of examples and many of them are just for those particular cases. That's what I mean in limitation of examples.

Now as for Shreve, I have learned Brownian motion, Ito's lemma and only small amount of stochastic processes while studding derivatives markets. Maybe this gave me the ease of learning it for now. But again, I have not found yet any special difficulty I couldn't handle. It is very common you searched for some models JIT you need it. For example I had to dig into some specific numerical integration methods while reading which doesn't at all mean I had to know it in advance before starting reading it. I am talking from my experience and not stating my opinions whether I am expecting it to be easy or difficult. Haven't yet finished the first part (about to finish in 1 week time). But in the second part's contents, the techniques are not that hell difficult.

One note: I agree with you in one opinion, calculus and statistics alone may not be enough to start learning it. But I didn't pay much attention to it since the basics of stochastic processes was familiar for me from other subjects. I have covered McDonald's derivatives markets and Hull's options, futures and other derivatives. So I might not have seen the specifics of those techniques making the books difficult to read. But the general opinion is the same for me, I am still a beginner in stochastic math but find it comfortable to learn on my background.
 
If you meant "Generating sigma-algebras" topic there, make sure you have read the topic till the end. The part you need is explained there.
 
Sometime in high school, during algebra class, the most interested I ever got was right before we started learning about compound interest and e. The teacher explained how it was first discovered and it's applications. I stayed up many nights trying to learning more and more about this strange new "mathematical constant" as we were learning it in school.

Somehow my story relates to your discussion about...
 
Sometime in high school, during algebra class, the most interested I ever got was right before we started learning about compound interest and e. The teacher explained how it was first discovered and it's applications. I stayed up many nights trying to learning more and more about this strange new "mathematical constant" as we were learning it in school.

Somehow my story relates to your discussion about...

e is not so complex to argue whether it is better to start with examples or analytical derivation. But good examples since getting the answer doesn't tell you which limit it is. My teacher also told the story about who discovered it with 18 digits precession. Explaining the concept of e with example first and then giving theoretical way is ok. All I say above is that theory is a good tool for introduction. Examples are auxiliary not vise versa.
 
BigBadwolf has the right idea. I have a bachelors in physics and I'm going through this book as we speak. The only financial background I have is from finishing Paul Wilmott's "Mathematics of Financial Derivatives", and I can honestly say that Shreve's book is quite difficult.

Since this book is so advanced, I have lifted my head out of it many times and looked through amazon for easier texts, but I always come back to Shreve. The way I see it is this; sure you could read a few easier books and buy Shreve when you feel more prepared, BUT if you're serious about becoming a quant you'll ABSOLUTELY have to know the material in this book. Therefore, you might as well grind it out and just have Google loaded up for whenever you don't fully grasp a concept (which is about every other page, seriously).
 
Since this book is so advanced, I have lifted my head out of it many times and looked through amazon for easier texts, but I always come back to Shreve.

There are other texts around that can give Shreve a run for his money but they seem not to known. For example, for Vol.1 of Shreve, there's van der Hoek and Elliott's "Binomial Models in Finance," published by Springer. It looks a bit more accessible than Shreve, and seemingly covers some extra topics. For Vol.2, there are quite a few rival treatments, many published by Springer Finance.
 
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