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I am a mathematician. What's the go-to reference for a proper math-based introduction to martingale theory and arbitrage pricing?

The books I am being referred to deal mostly either with the discrete case, or, if its continuous, then it does not contain all the proofs and there's a lot of hand-waving (for example, Bjork's Arbitrage theory in Continuous time, which does not even contain proper proofs of, say, the fundamental theorem of finance or the all-important Ito formula).

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As a mathematician I have preferred Karatzas and Shreve, "Brownian Motion and Stochastic Calculus" (ISBN 978-0387976556). It has all the theorems and proofs, and is well-written.

Shreve noted the popularity of his book and later wrote the 2-volume set "Stochastic Calculus for Finance". The second volume may also be satisfactory to you, but I have not read it.

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I don't think there many books that proof the fundamental theorem of asset pricing as is quite technical and not very interesting for the usual audience studying quantitative finance.

Also, Ito formula is stochastic calculus subject, is a requisite for many mathematical finance books.

That said, I like these books for a more formal approach:

For a proof of the fundamental theorems of asset pricing see:

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Regarding the proof of the Fundamental Theorems of Asset Pricing (FTAP), as explained by @FKaria not many books present exhaustively and rigorously the proof as it is quite long and technical while not that useful in practice. However, you might want to look at the paper that shows the result is the most general framework:

Delbaen, Freddy; Schachermayer, Walter (1994). "A General Version of the Fundamental Theorem of Asset Pricing". Mathematische Annalen. 300 (1): 463–520.

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