Markov properties

Suppose that $\mathcal{S}$ is a locally compact separable metric space and $C_{0}$ is the set of continuous functions on $\mathcal{S}$ that vanish at infinity. To say a continuous function $f$ vanishes at infinity means that given $\varepsilon >0$ there exists a compact set $K$ such that $\left|f(x)\right|<\varepsilon$ if $x \notin K$. Show that if Assumption 20.1 is replaced by the assumptions that $P_{t} f \in C_{0}$ whenever $f \in C_{0}$ and $P_{t} f \rightarrow f$ uniformly as $t \rightarrow 0$ whenever $f \in C_{0}$, then the conclusion of Theorem 20.9 still holds.

Suppose $\left(X_{t}, \mathbb {P}[x]\right)$ is a Markov process with respect to a filtration $\left\{ \mathcal{F}_{t}\right\}$. Suppose that $\mathcal{E}_{t} \subset \mathcal{F}_{t}$ for each $t$ and that $X_{t}$ is $\mathcal{E}_{t}$ measurable for each $t$. Show that $\left(X_{t}, \mathbb {P}[x]\right)$ is a Markov process with respect to the filtration $\left\{ \mathcal{E}_{t}\right\}$.

Give an example of a Markov process that is not a strong Markov process.

Hint: Let the state space be $[0, \infty )$ and starting from $x \in (0, \infty )$, let $X$ move deterministically at constant speed to the right. Starting at 0 , let $X$ wait an exponential length of time, and then begin moving at constant speed to the right.

Let $\left(X_{t}, \mathbb {P}[x]\right)$ be Brownian motion and let $\left\{ \mathcal{F}_{t}\right\}$ be the minimal augmented filtration. Suppose $B \in \vee_{t \geq 0} \mathcal{F}_{t}$ and for some $s>0$ is of the form $\; \mathbb {1}_{B}=\; \mathbb {1}_{A} \circ \theta_{s}$. Show that if $B$ is a $\mathbb {P}[x]$-null set for some $x$, then it is a $\mathbb {P}[x]$-null set for every $x$.

Let $P_{t}$ be transition probabilities for a Poisson process with parameter $\lambda$. These are defined in Exercise 19.3. Show that Assumption 20.1 holds.

Suppose $\left(X_{t}, \mathbb {P}[x]\right)$ is a Markov process with transition probabilities $P_{t}, f$ is a bounded Borel measurable function, $t_{0}>0$, and we define $M_{t}=P_{t_{0}-t} f\left(X_{t}\right)$ for $t \leq t_{0}$. Show that $\left(M_{t}, t \leq t_{0}\right)$ is a $\mathbb {P}[x]$-martingale for each $x$.

Use the Blumenthal $0-1$ law to show that if $W$ is a one-dimensional Brownian motion and $T=\inf \left\{ t>0: W_{t}>0\right\}$ is the first time Brownian motion hits $(0, \infty )$, then $\mathbb {P}\left(T=0\right)=1$.

Let $A$ be a Borel subset of a metric space $\mathcal{S}$. Let $T_{A}=\inf \left\{ t: X_{t} \in A\right\}$, where $\left(X_{t}, \mathbb {P}[x]\right)$ is a strong Markov process. Show that $\mathbb {P}_{x}\left(T_{A}=0\right)$ is either 0 or 1 for each $x$.

Let $\left(X_{t}, \mathbb {P}[x]\right)$ be a strong Markov process and let $A$ be a Borel subset of $\mathcal{S}$. We define $A^{r}$ by setting $A^{r}=\left\{ x: \mathbb {P}_{x}\left(T_{A}=0\right)=1\right\}$, where $T_{A}$ is the first hitting time of $A$. Thus $A^{r}$ is the set of points that are regular for $A$. Prove that for each $x$,