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Book for brushing up Maths


I am looking for a Maths book that could be good for getting the basics before starting a fin engg program . I have an engg background and have studied math for 3 semesters.
I am leaning towards "intro to stochastic processes " by sheldon ross and saw a few chapters online .... but due to no background on stochastic calculus , I am having diffciculty in understanding it . Kindly suggest whether I should go on and buy a hard copy of it or look into some other book for basics before jumping directly to it .

Wud highly appreciate any suggestions.
I know many will disagree but I am also new to stochastic calculus (currently learning) with Shreve's books. There are 2 volome books which cannot be regarded as introductory courses but you can easily take "Stochastic calculus and finance" which is combination of the materials split in those two ones. This is my subjective point. I donno if you find it easy or difficult but still, I'm currently learning it and find not that difficult (not very easy though).
thanks for the response . is there a book that is a combination of both the volumes ? Where can i find it ? Also, which volume do u think wud be better for starting - binomial asset pricing model or continuous time models ?
thanks for the response . is there a book that is a combination of both the volumes ? Where can i find it ? Also, which volume do u think wud be better for starting - binomial asset pricing model or continuous time models ?

First, Shreve's "Stochastic calculus and finance" is not a pure combination of those two volumes, it is just a draft which is a good introductory I think covering materials in a more slight way than the mentioned two volume books.
Second, you can't say any of them - binomial option pricing or continuous time models - is better to start with. The order is such:

Stochastic calculus and finance I - binomial option pricing
Stochastic calculus and finance II - Continuous time models

So if you embark on these books, why break sequence?!
It seems more logical to start with discrete models and then move on to the continuous but it is rather irrelevant right now since neither is regarded as introductory and requires highest knowledge of math.
Sheldon Ross's book "Stochastic Processes"

is not stochastic calculus. It is an intermediate-level book on stochastic processes, without any of the measure theory that you would need to understand Stochastic calculus, which is sometimes also referred to as Itō calculus, named for its inventor, Kiyoshi Itō .

Indeed, the first sentence of the publisher's description of Ross's book is

A nonmeasure theoretic introduction to stochastic processes.

If you are having trouble understanding this book by Sheldon Ross, then you should look at a more elementary book, for example "Introduction to Probability Models", also by Sheldon Ross.

And if that book is still too difficult, then you could try an even more elementary book, also by Sheldon Ross, called "A First Course in Probability."

While these books will provide you some background on elementary probability and on various stochastic processes, none of them will give you what you need to know to tackle the "really hard stuff" that makes up Stochastic Calculus. For this you also need to cover that part of mathematics called "Real Analysis", i.e., "Theory of Functions of a Real Variable." There are many books which address this topic, which most people find far more challenging than simple non-measure-theoretic probability and stochastic processes which are addressed by Ross's books.

One set of books which you might find quite helpful in getting the necessary background that you would need in order to understand the advanced stuff that you are interested in is a pair of books by Grimmett and Stirzaker. Their textbook is called "Probability and Random Processes" (be sure to get the third edition, from 2001) and their companion exercise book is called "One Thousand Exercises in Probability."

Another book which you might find helpful is "An Introduction to Measure-Theoretic Probability" by George Roussas.
Do not forget about the Linear Algebra - gilbert Strang's book is a good one.


If going with Strang for Linear Algebra, then instead of this one, I'd recommend Linear Algebra And Its Applications. Strang is brilliant teacher, but his books are infamous for its "conversational" style, which makes Introduction to Linear Algebra book rather unreadable (even if reader is following MIT 18.06 video lectures, that Strang is teaching following this book); I found Linear Algebra And Its Applications somewhat better in that regard.
I have one book of linear algebra explaining all of the contents with statistical examples. I liked it while reading some of the topics and my programming algorithms arose from that book. i don't remember the name of the author and I'll post when I get home. If you are going to learn linear algebra, this book is really brilliant. Thanks @cgorac for your suggestions too.
@myampol: thanks alot for the comprehensive insights. Cud u also throw on some insights on steven shreve's volumes ?

@Alexei: is prof Dan's book good for learning basics of stochastic calculus and prob ? or is it more regarding application of it to finance ?

@cgorac: thanks for the info ...
@achilles: prof Dan's book doesn't cover stochastic calculus, but it cover the application through Black Scholes equation. For Stochastic calculus you can take a look at Shreve's Stochastic Calculus for Finance II
@Alexei: is prof Dan's book good for learning basics of stochastic calculus and prob ? or is it more regarding application of it to finance ?
It has what one would be doing for the next year-two or more in the MFE program and it is a pre-requisite in some programs. It has both the theory and financial applications.
The Shreve's book is a good one too but it might not be the easiest one to start with; however, the knowledge of stochastic processes will give you a good head start when you will begin the actual course.

The following book is a good start with stochastic processes / probability theory:


Shreve's books are specific to financial mathematics. You would likely study these books once you are on an MFE program. While he does address many of the mathematical and statistical topics that you would need as background for stochastic calculus, some of this material is quite abstract and it is extremely challenging if you are seeing it for the first time.

It is very helpful to obtain some familiarity with various stochastic processes so that when you come to them in Shreve you are not seeing them for the first time. The probability material which I have mentioned above, particularly the Sheldon Ross books, are materials that you would likely use if you were studying an (applied) statistics degree and would help you understand various statistical concepts which are not always covered in a financial mathematics degree, where limitations of time dictate that certain topics cannot be covered. However, as I mentioned above, Ross omits anything requiring measure theory.

What really trips people up if they don't have a strong background in pure mathematics is the real analysis material. If, when you enter an MFE course, you have already obtained an understanding of things like sigma-algebras, filtrations, convergence, the Lebesgue integral (vs. the Riemann integral with which everyone else is familiar), the various flavors of limits and convergence (in probability, in distribution, almost-everywhere, almost-surely, etc.) and the other aspects of Measure Theory, then it would be much easier for you to understand the financial concepts which are covered in Shreve. You would definitely cover this material in the first year of any mathematics Ph.D. program, and it should also be covered in a pure mathematics MS degree. People who are not pursuing such degrees may never see this material.

While it is possible to learn it all at the same time, doing so is quite challenging. It is much easier to master Shreve if you are seeing these concepts for the second (or third) time, rather than if you are seeing everything for the first time.
Vol.1 of Shreve can be read without a background in real analysis or measure theory. Vol.2 is at a different level of complexity. It is wise to study the binomial model -- the discrete case -- first. The continuous model of Vol.2 arises as a limiting case (as delta T --> 0) of the binomial model. While in theory it is possible to study the continuous model first, from a pegagogical standpoint it is prudent to see the ideas in discrete form first.
BTW, there are quick review books for linear algebra...like

i just started viewing gilbert strang's video lectures on MIT OCW and am finding them quite useful .... will the 35 lectures on linear algebra take care of the course ?