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I am a senior undergraduate who has passed the first three actuarial exams on probability (P), financial mathematics (FM), and models for financial economics (MFE). I am working on passing the life contingencies exam in April.

For my final semester as an undergraduate, I am doing an independent study to learn about measure-theoretic probability theory in the context of actuarial science. I am especially interested in learning the theory behind Ito calculus and the proof of the Black-Scholes equation and formula, rather than just doing routine calculations using these formulas (like in exam MFE).

The current plan is to start off with A Probability Path and Adventures in Stochastic Proccesses both by Resnick, but none of these cover Ito calculus. They touch on Brownian motion and martingales for a little bit, but not very much of it.

I have taken two semesters of real analysis (we covered everything up to complete metric spaces and integration and differentiation in $\mathbb{R}^n$) and will be taking my second semester in abstract algebra (vector spaces, group actions, Sylow $p$-groups, and some other stuff I don't know about). Is there a text that we can use during this independent study that would be accessible to me that pertains to Ito calculus and its applications to finance (and/or actuarial science)?

Edit: Two texts that I have found in my research are Brzezniak and Zastawniak and Øksendal. Does anyone have a particular preference of one of these over the other? Are there any other texts you would recommend?

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The books you found are very good.

A classical is Steven Shreve's masterful two-volume text, Stochastic Calculus for Finance, which introduces students to stochastic calculus as a tool for financial derivative pricing

The celebrated books of L.C.G. Rogers, D. Williams, Diffusions, Markov processes, and martingales (my favourites)

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  • $\begingroup$ The Shreve looks intriguing. Do volumes I and II constitute one book, or is getting volume II sufficient? Rogers and Williams looks good as well... this will be a hard decision. $\endgroup$ – Clarinetist Jan 14 '14 at 22:33
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    $\begingroup$ You can get volume II only, the same for Rogers. If you are mone interested in financial application I suggest you Shreve; Rogers is more advanced from a mathematical point of view and has no many applications. $\endgroup$ – alexjo Jan 14 '14 at 22:48

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