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I am 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.

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Le Gall's "Brownian Motion, Martingales and Stochastic Calculus" may be a good fit. You need to be comfortable with functional analysis though, cause he uses it sometimes to streamline proofs.

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The best book in my opinion that gives the necessary background on stochastic calculus with sufficient details but without spending much time in technicalities is stochastic differential equations by oksendal. You can get to the essential (construction of Brownian motion and Ito Formula) in the first 5 chapters, but the whole book is a fun read.

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In my work I usually deal with Brownian motion and white noise so I am not the expert on the topic of fractional Brownian motion (fbm). But I recalled that I have seen an engineering application in the David Nualart paper on fbm (Section 4 - fbm in turbulence). Sorry, it is not on signal processing and it has a finance part (Section 5) but it is a good short intro on the topic of fbm. Also interesting paper for you could be this (fbm in a nutshell). You could check out the references in both of this papers additionaly.

But, if you type "fractional brownian motion" into the Science direct you get a lot of books that mention signal processing in engineering (fbm in engineering). And this would probably be your best shot in finding an answers to your questions.

Hope this helps you somehow.

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