See the edit below!
I do have a question about continuous-state branching processes while reading "Fluctuations of Lévy Processes with Applications" by Kyprianou:
Definition (Continuous-state branching process). A $[0, \infty]$-valued strong Markov process $Y=\{Y_t : t \geq 0\}$ with probabilities $ \{ P_x : x \geq 0 \}$ is called a continuous-state branching process if it has càdlàg paths and satisfies the branching property.
Let $Y$ be a continuous-state branching process. Then it is shown that the semi-group property $$ u_{t+s}(\theta) = u_t(u_s(\theta)) $$ is true where $u_r(\theta)$ fulfills the following equation: $$ E_x(e^{- \theta Y_r}) = e^{-xu_r(\theta)} \tag{$\star$} . $$ One step in the proof states $$ E_x(E(e^{-\theta Y_{t+s}}|Y_t)) = E_x(e^{-Y_t u_s(\theta)}). $$ I don't understand this step. What I tried to do was the following: $$ E(e^{-\theta Y_{t+s}}|Y_t) = E(e^{-\theta (Y_{t+s} - Y_t)} e^{-\theta Y_t}|Y_t) = E(e^{-\theta (Y_{t+s} - Y_t)}|Y_t) e^{-\theta Y_t} = E_{K}(e^{-\theta (\tilde{Y}_s - K)}|Y_t) e^{-\theta Y_t} $$ Here, I used in the second equality measurability. In the third equality, I want to use something of the form that $Y_{t+s} - Y_t$ is distributed like $\tilde{Y}_s$. But I don't want to start $\tilde{Y}$ at zero since such a branching process would then always be zero. As an alternative I thought of something like $\tilde{Y}_s - K$ where $\tilde{Y}$ is started at $K = Y_t$. However, I guess I am doing something wrong since I think we don't have any time homogeneity that I used and furthermore what I am doing with $K$ feels also weird but I don't know what else I should be doing since the process $Y_s$ is always zero if it is started at zero.
Assuming, however this is correct, I do can continue in the following way: $$ E_{K}(e^{-\theta (\tilde{Y}_s - K)}|Y_t) e^{-\theta Y_t} = E_{Y_t}(e^{-\theta \tilde{Y}_s}|Y_t) e^{\theta K} e^{-\theta Y_t} = e^{-\tilde{Y}_t u_s(\theta)} $$ which shows the desired step.
So my question is how this step is done properly and maybe also what my misconceptions are? Thanks a lot for your help.
Edit:
I was told that from the Markov property we can follow the equation $$ E(e^{-\theta Y_{t+s}}|Y_t) = E_{Y_t} (e^{-\theta Y_s}) $$ and using $(\star)$ the desired result follows.
However, I don't understand this step with the Markov property formally. Intuitively, it makes sense that the equation is true since if we know $Y_t$ (or $\mathcal{F}_t$ by the Markov property), the process has only time s to evolve from this, so this describes the same situation when would start a process at $Y_t$ and let it again evolve for time s. However, this sounds to me like some time homogeneity is needed and I don't see how this follows formally from the (strong) Markov property, i.e.: $$ P(Y_{t+s} \in B | \mathcal{F}_t) = P(Y_{t+s} \in B | Y_t). $$
So, I am wondering what I am missing/don't understand correctly in this situation. Thank you.