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I've in my studies taken (introductory, at the masters level) courses on both stochastic calculus, differential geometry (both elementary at the level of Pressley's book, and more advanced at the level of John Lee's "Introduction to Smooth Manifolds") and Riemannian geometry, all of which I have found very interesting.

However, I have also heard of there being an intersection of these subjects in what is apparently called stochastic differential geometry. Needless to say, that sounds incredibly fun, and I would like to try and study it in my free time.

The point of this post is, I would like some help in where to start. A quick Google-search brings up some articles, a couple sets of lecture notes, and some Springer books. I've had a quick look at those (previews in the case of books), but I'd like to have some established opinions to help me choose. Does anyone have any recommendations for someone learning the subject for the first time?

For the sake of comparison, in Algebra people often point to Dummit & Foote, Fraleigh, Lang, or maybe some other book when one asks for references. Topologists point to Munkres or Hatcher, differential geometers to Lee, Spivak or do Carmo, and in (real) analysis we have Baby-Rudin. I want to know if there are similar "universally acclaimed" references in stochastic differential geometry that I should look out for.

(I have seen the following post already, but I feel that I ask for a more general reference: what are prerequisite to study Stochastic differential geometry?)

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    $\begingroup$ Have you checked out Chirikjian's two titles on stochastic models, information theory and Lie groups? $\endgroup$ – J W Dec 29 '16 at 11:14
  • $\begingroup$ @JW I didn't know about these, thanks for the tip! $\endgroup$ – A.Sh Dec 29 '16 at 12:00
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I think the closest thing to what you are talking about is Stochastic Analysis on Manifolds by Hsu.

Many books (e.g. the ones mentioned in the comments above, or this one, which I like) have great sections on the topic, but are not only focused on it (I guess not enough people in the field :P).

Incidentally, you might also be interested in information geometry (e.g. see here), the study of probability distributions as Riemannian manifolds (kind of like an intersection of the subjects from a different viewpoint).

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