Rreference Request: book on stochastic calculus (not finance)

I am looking at looking at fractional Gaussian/Brownian noise from a signal theoretic and engineering point of view. In particular, I am looking at the math behind what defines these noise processes and what consequences this has on the physics, either generating them or consuming these noise signals.

As an engineer by training I am familiar with both (real/multivariate/complex) calculus and basic probability theory and also stochastic signals. But most of what I am doing now is where fractional calculus and stochastic calculus meet (Hic sunt dracones… literally). I think I can get my way around most of the fractional calculus part, but for the stochastic calculus I am in need of better understanding of how it works.

What I am looking for is a book (or lecture notes) that not only give me an understanding and intuition how stochastic calculus works (ie. how to apply it), but I also need the proofs in order to tell what I am allowed to do with the theorems and what not. Measure theory shouldn’t be much of a problem, as I have two mathematicians at hand who can explain things, if I get stuck.