Framework for Markov processes

Suppose $\left(X_{t}, \mathbb {P}[x]\right)$ is a Brownian motion and $S_{t}=\sup_{s \leq t} X_{s}$. Show that $\left(\left(X_{t}, S_{t}\right), \mathbb {P}[x]\right)$ is a Markov process and determine the transition probabilities.

Suppose $\left(X_{t}, \mathbb {P}[x]\right)$ is a Brownian motion, $f$ a non-negative, bounded, Borel measurable function, and $A_{t}=\int_{0}^{t} f\left(X_{s}\right) d s$. Show that $\left(\left(X_{t}, A_{t}\right), \mathbb {P}[x]\right)$ is a Markov process.

Suppose $P_{t}$ is a Poisson process with parameter $\lambda$. Let $\Omega^{\prime }$ be the collection of functions on $[0, \infty )$ which are right continuous and which have left limits, let $\mathcal{F}$ be the $\sigma$-field on $\Omega^{\prime }$ generated by the cylindrical subsets of $\Omega^{\prime }$, let $P_{t}^{x}=x+P_{t}$, and let $\mathbb {P}[x]$ be the law of $x+P$. Show that $\left(X_{t}, \mathbb {P}[x]\right)$ is a Markov process and determine the transition probabilities.

Suppose $m$ is a measure on the Borel subsets $\mathcal{B}$ of a metric space $\mathcal{S}$. Suppose for each $t>0$ there exist jointly measurable non-negative functions $p_{t}: \mathcal{S} \times \mathcal{S} \rightarrow \mathbb {R}$ such that $\int p_{t}(x, y) \, m(d y)=1$ for each $x$ and $t$ and define

Show that the kernels $P_{t}$ satisfy the Chapman-Kolmogorov equations if and only if

for every $s, t \geq 0$, every $x \in \mathcal{S}$, and $m$-almost every $z$.

The Ornstein-Uhlenbeck process $Y$ started at $x$ is a continuous Gaussian process with $\mathbb {E}\left[Y_{t}\right]=e^{-t / 2} x$ and covariance

If $X$ is the canonical process and $\mathbb {P}[x]$ is the law of an Ornstein-Uhlenbeck process started at $x$, show that $\left(X_{t}, \mathbb {P}[x]\right)$ is a Markov process and determine the transition probabilities.