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I searched for this question but couldn't find a direct answer. Basically I want to understand (and possibly compute some simple instances of) the Ito integral. I am coming from a physics background and have little experience with statistics. I encountered the idea in the topic of Quantum Path Integrals and I am now very curious.

So basically I am looking for a list of somewhat informal resources that would build me to the point of understanding and being able to numerically/analytically compute some simple Ito Integrals and understand what is going on.

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Perhaps you might find Jeffery S. Rosenthal's "A first look at rigorous probability theory" useful. It's a mathematically precise (but very readable) book which starts with basic measure theory and finishes with Ito diffusions and Brownian motion (although this section is relatively brief). However, the exercises are judiciously chosen and the author has the solutions to odd questions on his website, so it's quite nice for self-study.

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