What is a good stochastic calculus to read for a math master?

[Eric.Z]

Hello,

I have read some of the popular books that most math phds would read in their first 2 years of study. I am pretty good with analysis, measure theory, measure theoretic probability and an intro level stochastic calculus (Shreve's book II).

Particularly, the relevant books I have read include Royden's Real Analysis, Chung's book on probability, Shreve's book on Stochastic calculus for finance.

With this background, what is a good book for me to learn (teach myself) a rigorous treatment of stochastic calculus? Or should I gain more mathematical maturity before studying stochastic calculus?

 

[Daniel Duffy]

Most stochastic books are extremely intimidating unless you have an advanced degree in mathematics and even then it's not easy.

A benign intro is here

Elementary Stochastic Calculus With Finance in View (Advanced Series on Statistical Science & Applied Probability, Vol 6) (Advanced Series on Statistical Science and Applied Probability): Thomas Mikosch: 9789810235437: Amazon.com: Books

And then the practical numeric is here
Amazon.com: Numerical Solution of SDE Through Computer Experiments (Universitext) (9783540570745): Peter Eris Kloeden, Eckhard Platen, Henri Schurz: Books

And then some hands-on programing in your favourite language.

I suppose if you know all this stuff then the other book by Kloeden and Platen is good.

The book by Cont an Tankov is excellent.

And then the recent opus magnum of Alan Lewis should be mentioned.
Option Valuation under Stochastic Volatility II: With Mathematica Code: Alan L Lewis: 9780967637211: Amazon.com: Books

 

[Eric.Z]

hands

Thanks for your comments. I have heard of the book by Lewis and was told that the book was very hard. So if am determined to reach the level of this book, what is the sequence of subjects I should study? I was thinking maybe finish Stein's Book on Functional Analysis(the 4th one) and complete a course on PDE. Would that suffice?

 

[Daniel Duffy]

Thanks for your comments. I have heard of the book by Lewis and was told that the book was very hard. So if am determined to reach the level of this book, what is the sequence of subjects I should study? I was thinking maybe finish Stein's Book on Functional Analysis(the 4th one) and complete a course on PDE. Would that suffice?

You're welcome.

My take on the book.

Good question. A good knowledge of SDE is needed in any event. Also, probability densities, distributions, as well as some PDEs.

Some advanced formulae you can take at face value for the moment. You don't need to know Functional Analysis as such.

The nice thing is that most of the results are programmed in Mathematica. The author has projects, especially chaps 3, 5 and 9. So, you could take the formulae as requirements and then extend/apply them.

I suppose I can say it but I hope to get my MSc students theses to do some of these in C++ in the very short term.

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