Axioms of Brownian motion, filtrations and independence

Let $W_{t}$ be a Brownian motion and $\left\{ \mathcal{F}_{t}\right\}$ be the minimal augmented filtration generated by $W$. Let

Show that $T$ is not a stopping time with respect to $\left\{ \mathcal{F}_{t}\right\}$.

Let $X_{t}$ be a stochastic process and let $\left\{ \mathcal{F}_{t}^{00}\right\}$ be the filtration generated by $X$. Suppose $X$ is a Poisson process with respect to the filtration $\left\{ \mathcal{F}_{t}^{00}\right\}$. Show that $X$ is a Poisson process with respect to the minimal augmented filtration generated by $X$.

Hint: Imitate the proof of Proposition 2.5 .

Suppose $\left\{ \mathcal{F}_{t}\right\}$ is a filtration satisfying the usual conditions, $P_{t}$ is a Poisson process with respect to $\left\{ \mathcal{F}_{t}\right\}$, and $W_{t}$ is a Brownian motion with respect to $\left\{ \mathcal{F}_{t}\right\}$. Show that if $W_{t}+P_{t}$ has stationary and independent increments, then $P$ and $W$ are independent processes.

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 $\left\{ \mathcal{E}_{t}\right\}$.

Definition 7. A process $W = \left\{ W(t), t \geqslant 0\right\}$ is called a real Brownian motion with respect to a filtration $(\mathscr {F})_{t \geqslant 0}$ if

1. $W(t)$ is an $\mathscr {F}_{t}$-measurable variable for each $t \geqslant 0$;

2. $\left(W(t)-W(s)\right) \; \perp \! \! \! \perp \; \mathscr {F}_{s}$ for $0 \leqslant s < t$ (i.e. $W(t)-W(s)$ is independent of $\mathscr {F}_{s}$);

3. $W_{0} = 0$ a.s. and $W(t)-W(s) \sim \mathrm{N}(0, t-s)$ for $0 \leqslant s < t$;

4. the sample paths of the process $W$ are continuous a.s.

Let $\mathscr {N}$ be the class of events of zero probability of the (complete) probability space $(\Omega , \mathscr {F}, \mathbb {P})$. Without loss of generality we shall assume that in Definition 7 each $\sigma$-algebra $\mathscr {F}_{t}(t \geqslant 0)$ contains all events from $\mathscr {N}$.

Let $\left\{ \left(Y_{n}, Z_{n}\right)\right\}_{n \geqslant 1}$ be a sequence of independent random vectors with values in $\mathbb {R}^{2}$, and moreover $\left\{ \left(Y_{n}, Z_{n}\right)\right\}_{n \geqslant 1} \; \perp \! \! \! \perp \; W$, where $W = \left\{ W(t), t \geqslant 0\right\}$ is a Brownian motion (with respect to the filtration $\left(\mathscr {F}_{t}^{W}\right)_{t \geqslant 0}$ ). For $t \geqslant 0$ set $\mathscr {H}_{t} = \sigma \left\{ \mathscr {F}_{t}^{W}, \left\{ \left(Y_{n}, Z_{n}\right)\right\}_{n \geqslant 1}\right\}$, i.e. consider the $\sigma$-algebra generated by the variables $W(s)$ for $s \in [0, t]$ and $\left(Y_{n}, Z_{n}\right)_{n \geqslant 1}$. We shall assume it to be augmented by the class $\mathscr {N}$ of null events.

Show that $W$ is a Brownian motion with respect to the filtration $\left(\mathscr {H}_{t}\right)_{t \geqslant 0}$ introduced above.

For a filtration $\left(\mathscr {F}_{t}\right)_{t \geqslant 0}$ set $\mathscr {F}_{t+} = \bigcap_{s > t} \mathscr {F}_{s}$, $t \geqslant 0$.

Is it true that for the Wiener process and its natural filtration $\left(\mathscr {F}_{t}\right)_{t \geqslant 0}$ the relation $\mathscr {F}_{t} = \mathscr {F}_{t+}$ holds for all $t \geqslant 0$ ?

Wiener process in $n$ dimensions. An $n$-dimensional Wiener process is a process $W(t) = \left(W_{1}(t), W_{2}(t), \ldots , W_{n}(t)\right)$ taking values in $\mathbb {R}^{n}$ with independent increments, started at the origin $W(0) = 0$, and such that $W(t)-W(s)$ is multivariate normal with means 0 and covariance matrix $(t-s) \mathbf{I}$ where $\mathbf{I}$ is the $n \times n$ identity matrix. Let $W(t)$ be an $n$-dimensional Wiener process, and let $A$ be an orthonormal $n \times n$ matrix. Show that $A W(t)$ is an $n$-dimensional Wiener process.

Let $\left(X_{t}, t \geq 0\right)$ be a Gaussian process with stationary independent increments. Prove that there exist constants $a, b \in \mathbb {R}$ such that the process

is a Wiener process.

Consider the condition for all $s < t$ the random variables $\frac{B(t) - B(s)}{\sqrt{t - s}}$ are identically distributed, centred and have variance 1.

Show that (B0), (B1), (B2), (B3) and (B0), (B1), (B2), (B3') are equivalent.

Let $\left(X_{t}\right)_{t \geqslant 0}$ be a stochastic process on the measure space $(\Omega , \mathscr {A}, \mathbb {P})$ satisfying (B1)(B3). Assume further that there is some measurable set $\Omega_{0} \subset \Omega , \mathbb {P}\left(\Omega_{0}\right) = 1$, such that for all $\omega \in \Omega_{0}$ we have $X_{0}(\omega ) = 0$ and $t \mapsto X_{t}(\omega )$ is continuous. Show that

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$. Show that $\mathscr {F}_{\infty }^{B} = \bigcup_{J \text{ countable }}^{J \subset [0, \infty )} \sigma (B(t): t \in J)$.

Show that (B1) is equivalent to $B_{t} - B_{s} \; \perp \! \! \! \perp \; \mathscr {F}_{s}^{B}$ for all $0 \leqslant s < t$.

Find out whether the processes $X(t): = B\left(e^{t}\right)$ and $X(t): = e^{ - t / 2} B\left(e^{t}\right), t \geqslant 0$, have the no - memory property, i.e. $\sigma (X(t): t \leqslant a) \; \perp \! \! \! \perp \; \sigma (X(t+a) - X(a): t \geqslant 0)$ for $a > 0$.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{d}$ and assume that $X$ is a $d$ - dimensional random variable which is independent of $\mathscr {F}_{\infty }^{B}$.

Let $\left(\mathscr {F}_{t}\right)_{t \geqslant 0}$ be an admissible filtration for the Brownian motion $\left(B_{t}\right)_{t \geqslant 0}$. Mimic the proof of Lemma 2.14 and show that $\mathscr {F}_{t} \; \perp \! \! \! \perp \; \mathscr {F}_{[t, \infty )}^{W}: = \sigma \left(B_{u} - B_{t}: u \geqslant t\right)$ for each $t > 0$.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and $\sigma , \tau$ be two stopping times such that $\mathbb {E} \tau , \mathbb {E} \sigma < \infty$.

In the proof of Theorem 14.2 we assume that $\left(B_{t}\right)_{t \geqslant 0}$ and $(U, W)$ are independent. Show that $\mathscr {F}_{t}: = \sigma \left(B_{s}, s \leqslant t ; U, W\right)$ is an admissible filtration for a Brownian motion, cf. Definition 5.1.