Brownian motion as a Markov process

Let $\left(B_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$.

Compare the (last part of the) first ("pedestrian") proof of Theorem 6.5 with the characterization of a BM by Lemma 5.4. Can you think of a way to use this lemma instead of the direct calculation?

Assume that $\left(X_{t}\right)_{t \geqslant 0}$ is a uniformly bounded stochastic process with exclusively continuous sample paths, $\left(\mathscr {F}_{t}\right)_{t \geqslant 0}$ is some filtration, and $\sigma$ is an everywhere finite stopping time.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a Brownian motion on the space $\left(\mathcal{C}, \mathscr {B}(\mathcal{C}), \mathbb {P}^{0}\right)$. The (canonical) shift operator is the map $\theta_{h}: \mathcal{C} \rightarrow \mathcal{C}, \theta_{h} w(\cdot ) = w(\cdot +h)$, i.e. $\left(\theta_{h} w\right)(t) = w(t+h)$. Moreover, define $\mathscr {F}_{t} = \mathscr {F}_{[0, t]} = \sigma \left(B_{s}, s \in [0, t]\right)$ and $\mathscr {F}_{[t, \infty )} = \sigma \left(B_{s}, s \geqslant t\right), \quad t \in [0, \infty ]$.

If $\sigma$ is a stopping time for $\left(\mathscr {F}_{t}\right)_{t \geqslant 0}$ which is finite for all $w$, the random shift is defined as $\left(\theta_{\sigma } w\right)(t): = \left(\theta_{\sigma (w)} w\right)(t) = w(t+\sigma (w))$.

Let $\left(B_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and set $M_{t}: = \sup_{s \leqslant t} B_{s}$ and $I_{t} = \int_{0}^{t} B_{s} d s$.

Let $(B(t))_{t \geqslant 0}$ be a $\mathrm{BM}^{d}$ and let $Z$ be a bounded $\mathscr {F}_{\infty }^{B}$ measurable random variable. Then $x \mapsto \mathbb {E}^{x} Z$ is in $\mathcal{B}_{b}\left(\mathbb {R}^{d}\right)$.

Let $\left(B_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ such that $\mathscr {F}_{0}$ contains all measurable null sets and let $\tau$ be a stopping time. Show that $B_{\tau }$ is measurable with respect to $\mathscr {F}_{\tau +}$.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$. Show that $\mathbb {P}\left(\sup_{s \leqslant t}\left|B_{s}\right| \geqslant x\right) \leqslant 2 \mathbb {P}\left(\left|B_{t}\right| \geqslant x\right), x, t \geqslant 0$.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and denote by $\tau_{b} = \inf \left\{ s \geqslant 0: B_{s} = b\right\}$ the first time when $B_{t}$ reaches $b \in \mathbb {R}$. Show that

Let $\tau = \tau_{(a, b)^{c}}^{\circ }$ be the first exit time of a Brownian motion from the interval $(a, b)$.

Let $W = \left(W_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{2}$ such that $W_{0} = (a, b)$ with $a, b > 0$. What is the probability that $W_{t}$ hits first the positive part of the $x$ - axis before it hits the negative part?

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and set $M_{t}: = \sup_{s \leqslant t} B_{s}$. Find the distribution of $\left(M_{t}, B_{t}\right)$.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and let $\tau_{0}$ be the first hitting time of 0. Find the "density" of $\mathbb {P}^{x}\left(B_{t} \in d z, \tau_{0} > t\right)$, i.e. find the function $f_{t, x}(z)$ such that

Let $K \subset \mathbb {R}^{d}$ be a compact set. Show that there is a decreasing sequence of continuous functions $\phi_{n}(x)$ such that $\; \mathbb {1}_{K} = \inf_{n} \phi_{n}$.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$. Find the distribution of $\widetilde{\xi }_{t}: = \inf \left\{ s \geqslant t: B_{s} = 0\right\}$.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and set $M_{t} = \sup_{s \leqslant t} B_{s}$. Denote by $\xi_{t}$ the largest zero of $B_{s}$ before time $t$ and by $\eta_{t}$ the largest zero of $Y_{S} = M_{s} - B_{s}$ before time $t$. Show that $\xi_{t} \sim \eta_{t}$.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and $0 < u < v < w < \infty$. Find the following probabilities: