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Thanks to many people who have mentioned it to me and others on this site before. I was just able to peek into Kallenberg' Foundation of Modern Probability. It is more comprehensive, deep and thorough than the books I have seen before. It not only covers probability theory, but also stochastic processes and calculus, random measures, point processes and other topics. I don't expect myself to understand it fully and easily at all, but I admit it is the best reference I have seen.

I was wondering if you could mention other books at similar (partially) topics and/or levels as Kallenberg' book?

Thanks and regards!

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"Probability With a View Towards Statistics" in two volumes, by J. Hoffman-Jorgensen. –  Michael Greinecker Feb 17 '13 at 20:11
@MichaelGreinecker: Thanks! I remember I heard that book from you or someone else a while ago. But I haven't been able to find an electronic copy. –  Tim Feb 17 '13 at 20:19
I've only seen a physical copy of that book. There is also a classic in advanced probability, "Probabilities and potential" by Dellacherie and Meyer, which comes in three volumes. –  Michael Greinecker Feb 17 '13 at 20:27
@Tim : I can cite the book of Shiryaev and Jacod "Limit theorems for stochastic Process" which IMO a monument. But with high pre requisit on theory of stochastic processes and probability. –  TheBridge Feb 17 '13 at 22:35
@TheBridge: Thanks, that one is great! –  Tim Feb 17 '13 at 22:59

1 Answer 1

One very useful reference is Rogers & Williams: "Diffusions, Markov processes and Martingales". It's not quite as varied in its topics as Kallenberg's book, but it does cover many topics in the theory of stochastic processes - both discrete-time and continuous-time martingales, weak convergence on Polish spaces, regular conditional probabilities, Markov processes and stochastic integration.

The book "Semimartingale theory and stochastic calculus" by He, Wang & Yan is also rather useful when it comes to continuous-time stochastic processes. It covers both parts of the classical "general theory of processes" as is also found in Meyer & Dellacherie, as well as a very good deal of martingale theory, stochastic integration with respect to general semimartingales, and also Girsanov's theorem, martingale representation, various inequalities, random measures and weak convergence of semimartingales.

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