I'm kind of struggling with jumping behavior of some basic processes coming from the Doob-Meyer decomposition. Unfortunately I couldn't find a book that elaborates enough on this, but if anybody knows one, this would be tremendously helpful!
Let $M$ be a càdlàg $\mathcal{F}_t$-martingale and $\tau$ be a predictable stopping time. Is it always true, that $$ E[\Delta M(\tau)]=E[E[\Delta M(\tau)|\mathcal{F}(\tau-)]]=0? $$ Why does it hold true and where can I find more on this topic?
Also: If $A$ is the predictable compensator coming from the Doob-Meyer decomposition of a càdlàg submartignale and $\tau^{A}$ is it's first jump, is $\tau^{A}$ predictable?
Thanks in advance!