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I am an aspiring quant that would like to get a head start learning stochastic calculus, which books FROM EXPERIENCE are the most reader friendly?

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You can also look in bjorks book Arbitrage Theory in continous time which is relatively comprehensive and not too technical. On the downside he is not very consistent with his notations inbetween chapters (it is as they are written at different times throughout his career and then mashed together). –  Good Guy Mike Apr 29 at 15:09

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For a basic introduction, the three chapters in Hull's Options, Futures, and Other Derivatives on Binomial Trees, Wiener Processes and Ito's Lemma, and The Black-Scholes-Merton Model helped me start to understand the basic concepts within a broader context.

After that, Shreve's two books seems to be pretty popular (see here and here). He explains things pretty thoroughly in my opinion and his writing is pretty clear, which I like.

For someone starting out wanting to learn on their own, though, I've heard his books can be a bit tough to wade through. If you find that to be the case, check out Mikosch's Elementary Stochastic Calculus With Finance in View.

Another one that might be helpful is Basic Stochastic Processes by Brzezniak and Zastawniak. It reviews basic probability and other fundamentals, and has solved exercises, which could be really nice for someone starting off and learning on their own.

And then another one I've seen mentioned is Mathematical Methods for Financial Markets. I haven't read it personally, but judging from the Table of Contents it seems to cover a good breadth of advanced topics if that's more of interest to you.

Which one you go with kind of depends on your experience, what math/stats you've taken, your personal preferences for style and writing/math balance so to speak. But hopefully this gives you enough to get your started :)

(Oh, and I also came across this page from an NYU class on partial differential equations for finance. It has PDF lecture notes that might be helpful, depending on what you're looking for. Specifically, the Lecture 1 notes start off with a discussion of the "Links between stochastic differential equations and PDE.")

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