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Let

  • $(\Omega,\mathcal A,\operatorname P)$ be a probability space
  • $(E,\mathcal E)$ be a measurable space
  • $(Y_n)_{n\in\mathbb N_0}$ be a $(E,\mathcal E)$-valued time-homogeneous Markov chain on $(\Omega,\mathcal A,\operatorname P)$ with transition kernel $\kappa$
  • $(N_t)_{t\ge0}$ be a Poisson process on $(\Omega,\mathcal A,\operatorname P)$ with intensity $\lambda\in[0,\infty)$
  • $B(E,\mathcal E):=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$ be equipped with $$\left\|f\right\|_\infty:=\sup_{x\in E}\left|f(x)\right|\;\;\;\text{for }f:E\to\mathbb R$$

Assume $Y$ and $N$ are independent. Let $$X_t:=Y_{N_t}\;\;\;\text{for }t\ge0.$$ We may identify $\kappa$ with a contractive and nonnegative linear operator on $B(E,\mathcal E)$ via $$Bf:=\int\kappa(\;\cdot\;,{\rm d}y)f(y)\;\;\;\text{for }f\in B(E,\mathcal E).\tag1$$ Now, let $$A:=\lambda\left(B-\operatorname{id}_{B(E,\:\mathcal E)}\right).$$ Note that $A$ is the infinitesimal generator of the contractive semigroup $$T(t):=e^{-\lambda t}e^{\lambda tB}\;\;\;\text{for }t\ge0.$$

How do we need to define a filtration $(\mathcal G_t)_{t\ge0}$ on $(\Omega,\mathcal A)$ such that $X$ is a time-homogeneous $\mathcal G$-Markov process on $(\Omega,\mathcal A,\operatorname P)$ whose transition semigroup corresponds to $\left(T(t)\right)_{t\ge0}$?

In the book of Ethier and Kurtz, the authors define $$\mathcal F_t:=\sigma(\sigma(N_s:s\le t)\cup\sigma(X_s:s\le t))\;\;\;\text{for }t\ge0.$$ Now, letting $s,t\ge0$ and $f\in B(E,\mathcal E)$, they claim that \begin{equation}\begin{split}\operatorname E\left[f(X_{t+s})\mid\mathcal F_t\right]&=\sum_{n=0}^\infty\operatorname E\left[1_{\left\{\:N_{t+s}\:-\:N_t\:=\:n\:\right\}}f\left(Y_{N_t\:+\:n}\right)\mid\mathcal F_t\right]\\&=\sum_{n=0}^\infty1_{\left\{\:N_{t+s}\:-\:N_t\:=\:n\:\right\}}\operatorname E\left[f\left(Y_{N_t\:+\:n}\right)\mid\mathcal F_t\right].\end{split}\tag1\end{equation}

So, they seem to use that $1_{\left\{\:N_{t+s}\:-\:N_t\:=\:n\:\right\}}$ is independent of $\sigma\left(f\left(Y_{N_t\:+\:n}\right),\mathcal F_t\right)$ for all $n\in\mathbb N_0$. While it's clear by the definition of a Poisson process that $N_{t+s}-N_t$ is independent from $N_t$, I don't get why it's even independent of $\mathcal F_t$.

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