The way our script and the books I found introduce martingale compensators is not intuitive to me and I would appreciate any help in understanding it:
Let $(M_n)_{n \in \mathbb{N}}$ be a martingale w.r.t. filtration $(\mathfrak{F}_n)_{n \in \mathbb{N}}$ with $E[M_n^2] < \infty$. Then $(M_n^2)_{n \in \mathbb{N}}$ is a submartingale and there exists a growing predictable process $(\langle M \rangle_n)_{n \in \mathbb{N}}$ such that $(M_n^2 - \langle M \rangle_n)_{n \in \mathbb{N}}$ is a martingale. The process $(\langle M \rangle_n)_{n \in \mathbb{N}}$ is called compensator of $(M_n)_{n \in \mathbb{N}}$ and we have:
$E[(M_n - M_{n-1})^2| \mathfrak{F}_{n-1}] = \langle M \rangle_n - \langle M \rangle_{n-1}$
I have read the excellent posts on the intuition of $\sigma$-Algebras, conditional expectations and martingales but I could not find one tackling this topic. It would help a lot to understand having an answer to these questions:
a) Are there recurring/important examples of compensators?
b) What applications are they used for?
c) How does the compensator relate to the intuition of martingales as a fair game? Or if they don't, how else can one understand the concept?
Would be great to get an answer to any of these questions. References to books that offer good explanations on the matter are also appreciated.