I'm studying the book "Arbitrage Theory in Continuous Time" by Bjork, and the authore uses a lot the notation $E^Q_{t,z}$, where $Q$ is a probability measure and $z=Z_t$ a stochastic process, but he did not give the definition.
Form the exercise below (taken from the book) I guess that $E^Q_{t,z}(Z_u)=E^Q(Z_u|Z_t)$. So for $u = t$ we should have $E^Q_{t,z}(Z_u)=Z_t$, is this correct?
The notation $\mathbb E^Q_{s,a}\big[B_t\big]$, where $B$ a stochastic process, $Q$ is the probability measure, $s\in\mathbb R^+$ a given time, $a\in\mathbb R$ a given value for the process $B$, is a shortcut for the longer notation for the conditional expectation of the stochastic process $\mathbb E^Q\big[B_t|B_s=a\big]$.
If the two times coincide, i.e. $t\equiv s$, then the result is: $$\mathbb E^Q_{s,a}\big[B_s\big]= \mathbb E^Q\big[B_s|B_s=a\big]=a.$$