Why does a time-homogeneous Markov process possess the Markov property?
Klenke defines (Definition 17.3, p. 346) a time-homogeneous Markov process independently, rather than as a special case of a stochastic process that possesses the Markov property (Definition 17.1, p. 345) and in addition satisfies certain further constraints. He then goes on to claim that "a time-homogeneous Markov process is simply a stochastic process with the Markov property and for which the transition probabilities are time-homogeneous" (Remark 17.4, p. 346).
How is the Markov property implied by the definition of a time-homogeneous Markov process?
Relevant Definitions
Setting Let $I\subseteq \left[0,\infty\right)$ and $\left(X_t\right)_{t\in I}$ be an $\mathbb{R}^n$-valued stochastic process vis-a-vis $\left(\mathcal{F}_t\right)_{t\in I}$, the natural filtration generated by $\left(X_t\right)_{t\in I}$.
Definition 17.1 We say that $X$ has the Markov property (MP) iff for every $A\in\mathcal{B}\left(\mathbb{R}^n\right)$ and all $s,t\in I$ with $s\leq t$, $$\mathrm{P}\left[\left.X_t\in A\space\right|\mathcal{F}_s\right]=\mathrm{P}\left[\left.X_t\in A\space\right|X_s\right]$$
Definition 17.3 Let $I$ be closed under addition and assume $0\in I$. A stochastic process $X = \left(X_t\right)_{t\in I}$ is called a time-homogeneous Markov process with distributions $\left(\mathrm{P}_x\right)_{x\in \mathbb{R}^n}$ on the space $\left(\Omega,\mathcal{A}\right)$ iff:
- For every $x\in\mathbb{R}^n$, $X$ is a stochastic process on the probability space $\left(\Omega,\mathcal{A},\mathrm{P}_x\right)$ with $\mathrm{P}_x\left[X_0=x\right]=1$.
- The map $\kappa:\mathbb{R}^n\times\mathcal{B}\left(\mathbb{R}^n\right)^{\otimes I}\rightarrow \left[0,1\right]$, $\left(x,B\right)\mapsto\mathrm{P}_x\left[X\in B\right]$ is a stochastic kernel.
- $X$ has the time-homogeneous Markov property: For every $A\in\mathcal{B}\left(\mathbb{R}^n\right)$, every $x\in\mathbb{R}^n$ and all $s,t\in I$, we have $$\mathrm{P}_x\left[\left.X_{t+s}\in A\space\right|\mathcal{F}_s\right]=\kappa_t\left(X_s,A\right)\space\space\mathrm{P}_x\mathrm{-a.s.}$$ Here, for every $t\in I$, the transition kernel $\kappa_t:\mathbb{R}^n\times\mathcal{B}\left(\mathbb{R}^n\right)\rightarrow\left[0,1\right]$ is the stochastic kernel defined for $x\in \mathbb{R}^n$ and $A\in\mathcal{B}\left(\mathbb{R}^n\right)$ by $$\kappa_t\left(x,A\right):=\kappa\left(x,\left\{y\in \mathbb{R}^I:y\left(t\right)\in A\right\}\right)=\mathrm{P}_x\left[X_t\in A\right]$$
The family $\left(\kappa_t\left(x,A\right),\space t\in I,x\in\mathbb{R}^n, A\in\mathcal{B}\left(\mathbb{R}^n\right)\right)$ is also called the family of transition probabilities of $X$.
Since $X_s$ is $\mathcal F_s$-measurable, the tower property shows that $$ \mathrm{P}_x\left[\left.X_{t+s}\in A\space\right|X_s\right]=\mathrm E_x\left[Y\left.\space\right|X_s\right],\qquad \text{where}\quad Y=\mathrm{P}_x\left[\left.X_{t+s}\in A\space\right|\mathcal{F}_s\right]. $$ Item (3) of Definition 17.3 asserts that $Y$ is $\sigma(X_s)$-measurable, hence $$ \mathrm E_x\left[Y\left.\space\right|X_s\right]=Y, $$ and, in particular, the property in Definition 17.1 holds.