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More mathematics than Steven Shreve's book

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Joined
6/11/10
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189
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Hello Everyone,

I am looking for some books on the foundations of Stochastic Calculus with more mathematics. Because I feel even Shreve's book isn't clear enough on some topics. What makes it worse is that some French book like Mathematical Methods for Financial Markets seems so difficult. I need a bridge between Shreve's application and the solid mathematics.

Do you happen to know one textbook as this bridge? Thanks.
 
Have you tried 'SDE: an intro with application' of Oksendal?
 
It seems very short. Haven't tried yet.
Currently I have downloaded these books:

Arbitrage Theory in Continuous Time by Bjork
Mathematics of Financial Market by Elliott and Kopp
_They are at the same level as Shreve

Continuous Martingales with Brownian Motion by Marc Yor - difficult
Measures, Integrals and Martingales by Rene Schilling - intense
A Probability Path & Adventures in Stochastic Processes by Resnik - I regret buying them because they are too long to study
Basic Stochastic Processes by Brzezniak & Ziastawniak - Names are difficult but content easy to follow, too brief though
Continuous Stochastic Calculus by Meyer - a good choice
Introduction to Stochastic Calculus with Applications by Fima Klebaner - a very good choice
 
what is that you are looking for. I would say forget books and ask yourself what is your background. have you taken courses in measure theory or pde. stochastic calculus assumes knowledge of probability.

surprising you say that Basic Stochastic Processes by Brzezniak & Ziastawniak is too brief.

I would add probability with martingales and A Probability Path & Adventures in Stochastic Processes by Resnik. You need a good knowledge of probability and pdes. And depends on how much time.

Michael Steele has a stochastic calculus.

Springer has a few more. what is the end purpose of your study.
 
Why u need so many books in Stoch Calc? If u really know Probability, PDE, then you can read 2-3 books in Stoch and that will be fine. After that u can read specific papers in specific field that you are interested in.
 
martingaletrader said:
what is that you are looking for. I would say forget books and ask yourself what is your background. have you taken courses in measure theory or pde. stochastic calculus assumes knowledge of probability.

surprising you say that Basic Stochastic Processes by Brzezniak & Ziastawniak is too brief.

I would add probability with martingales and A Probability Path & Adventures in Stochastic Processes by Resnik. You need a good knowledge of probability and pdes. And depends on how much time.

Michael Steele has a stochastic calculus.

Springer has a few more. what is the end purpose of your study.
Click to expand...

I lack measure and probability background. The only thing I know about is statistics at undergraduate level. I do have
Models for Probability and Statistical Inference by Stapleton - have not covered convergence part
However its focus is on Classical probability instead of stochastic probability processes.


I found Klebaner's book answering my hitting time questions, plus it looks like Shreve's Calculus but is deeper. I call it Explanation on Shreve.

Basic Stochastic Processes by Brzezniak & Ziastawniak might be a good probability book to start with? for somebody like me who likes measure and probability convergence knowledge?

Only then can I read Resnik? So Brzezniak & Ziastawniak+Resnik = expertise in stochastic probability?

Besides, I bought Fourier Methods in Finance for series solution on B-S PDE.
 
DVV said:
Why u need so many books in Stoch Calc? If u really know Probability, PDE, then you can read 2-3 books in Stoch and that will be fine. After that u can read specific papers in specific field that you are interested in.
Click to expand...

The reason is that Shreve's book does not provide sufficient knowledge for me to read papers.
 
I guess the deeper mathematics than Shreve involve mainly two areas:
One is deeper Calculus like Protter
One is deeper probability with martingales like Williams

And do I need to understand both?
 
One book that I really enjoy reading is Martingale Methods in Financial Modelling by Musiela & Rutkowsky is very rigurous with mathematics developement and you don't need a lot of Stochastic Calculus to follow it.

My knowdeledge of Stochastic Calculus is reduced to a subject I took during my Advanced Mathematics Master in Barcelona from Marta San'z-Solé. Lecture notes are avaliable on her webpage.
 
martingaletrader and DVV are right!

Background in Stat is good but there are still a lot to know to make a solid background.
regarding your first post; you said that you need more math. but with your background, more math will work for you only when you have a good intuition about the game you want to start, that is, stochastic calculus. In other words, if you know, intuitively, what the aim of stochastic calculus is, or why it was invented, the stories around Brownian motion and etc, then you can attack more math even if you don't have background in Prob theory or PDE or ...;
anyway, I think Bjork's book is a thorough introduction with both math and good explanation.
after all, it depends what you want out of stochastic calculus. writing only a report ...or want to start a graduate study; if the latter, then i recommend don't rush and take small steps.
good luck
 
FKaria said:
One book that I really enjoy reading is Martingale Methods in Financial Modelling by Musiela & Rutkowsky is very rigurous with mathematics developement and you don't need a lot of Stochastic Calculus to follow it.

My knowdeledge of Stochastic Calculus is reduced to a subject I took during my Advanced Mathematics Master in Barcelona from Marta San'z-Solé. Lecture notes are avaliable on her webpage.
Click to expand...
Thank you. I have downloaded her lecture notes.
 
amir said:
martingaletrader and DVV are right!

Background in Stat is good but there are still a lot to know to make a solid background.
regarding your first post; you said that you need more math. but with your background, more math will work for you only when you have a good intuition about the game you want to start, that is, stochastic calculus. In other words, if you know, intuitively, what the aim of stochastic calculus is, or why it was invented, the stories around Brownian motion and etc, then you can attack more math even if you don't have background in Prob theory or PDE or ...;
anyway, I think Bjork's book is a thorough introduction with both math and good explanation.
after all, it depends what you want out of stochastic calculus. writing only a report ...or want to start a graduate study; if the latter, then i recommend don't rush and take small steps.
good luck
Click to expand...

Currently my research focuses on double barriers, a rare topic in textbooks.
 
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