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.
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 probabili…
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.
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 inter…
