The canonical model

Let $F: \mathbb {R} \rightarrow [0,1]$ be a distribution function.

Show that there exists a stochastic process $\left(X_{t}\right)_{t \geqslant 0}$ such that the random variables $X_{t}$ are independent $\mathrm{N}(0, t)$ random variables. This process also satisfies

Let $S$ be any set and $\mathscr {E}$ be a family of subsets of $S$. Show that

Let $\emptyset \neq E \in \mathscr {B}\left(\mathbb {R}^{d}\right)$ contain at least two elements, and $\emptyset \neq I \subset [0, \infty )$ be any index set. We denote by $E^{I}$ the product space $E^{I} = \left\{ f: f: I \rightarrow E\right\}$ which we equip with its natural product topology (i.e. $f_{n} \rightarrow f$ if, and only if, $f_{n}(t) \rightarrow f(t)$ for every $t \in I)$. Then we can define the Borel $\sigma$ - algebra $\mathscr {B}\left(E^{I}\right)$ and the product $\sigma$ - algebra $\mathscr {B}^{I}(E): = \sigma \left(\pi_{t}: t \in I\right)$ where $\pi_{t}: E^{I} \rightarrow E, \pi_{t}(f): = f(t)$ is the canonical projection onto the $t$ th coordinate.

Let $\mu$ be a probability measure on $\left(\mathbb {R}^{d}, \mathscr {B}\left(\mathbb {R}^{d}\right)\right), v \in \mathbb {R}^{d}$ and consider the $d$ dimensional stochastic process $X_{t}(\omega ): = \omega +t v, t \geqslant 0$, on the probability space $\left(\mathbb {R}^{d}, \mathscr {B}\left(\mathbb {R}^{d}\right), \mu \right)$. Give an interpretation of $X_{t}$ if $\mu = \delta_{x}$ for some $x \in \mathbb {R}^{d}$ and determine the family of finite dimensional distributions of $\left(X_{t}\right)_{t \geqslant 0}$.

Let $B = \left(B_{t}\right)_{t \in [0,1]}$ be a $\mathrm{BM}^{1}$ and set $W = \left(W_{t}\right)_{t \in [0,1]}$ where $W_{t}: = B_{t} - t B_{1}$.

Let $\left(\mathcal{C}_{(\mathrm{o})}, \mathscr {B}\left(\mathcal{C}_{(\mathrm{o})}\right), \mu \right)$ be the canonical Wiener space (we assume $d = 1$ ). We will now consider the space $\mathcal{C}$, again equipped with the metric of locally uniform convergence and the Borel $\sigma$ - algebra $\mathscr {B}(\mathcal{C})$. It is straightforward to check that

Lemma 4.1 remains valid in this context. We define $B_{t}: \mathcal{C} \rightarrow \mathbb {R}$ as coordinate projection $B_{t}(w) = w(t)$ where $w \in \mathcal{C}$ and the corresponding $\sigma$ - algebras

Let $h > 0$. The (canonical) shift operator is the map $\theta_{h}: \mathcal{C} \rightarrow \mathcal{C}, \theta_{h} w(\cdot ) = w(\cdot +h)$, i.e. $\left(\theta_{h} w\right)(t) = w(t+h)$.

We continue with the set - up introduced in Problem 7. Denote by $\left(\mathcal{C}_{(0)}, \mathscr {B}\left(\mathcal{C}_{(0)}\right), \mu \right)$ the canonical Wiener space and set $\Omega = \mathcal{C}, \mathscr {A} = \mathscr {B}(\mathcal{C})$, $B_{t}(w) = w(t)$ for $w \in \Omega$, and define for every probability measure $\pi$ on $(\mathbb {R}, \mathscr {B}(\mathbb {R}))$ a probability measure $\mathbb {P}^{\pi }$ on $(\Omega , \mathscr {B}(\mathcal{C})$ ) first on the "cylinder sets"

where $n \in \mathbb {N}_{0}, t_{0} = 0 < t_{1} < \cdots < t_{n}$ and $C_{0}, C_{1}, \ldots , C_{n} \in \mathscr {B}(\mathbb {R})$. The measure $\mathbb {P}$ is then extended, by Kolmogorov's theorem (Theorem 4.8), onto $\mathscr {B}(\mathcal{C})$.

Let $B = \left(B_{t}\right)_{t \geqslant 0}$ be a canonical $\mathrm{BM}^{1}$ on Wiener space $(\Omega , \mathscr {A}, \mathbb {P}) = \left(\mathcal{C}_{(0)}, \mathscr {B}\left(\mathcal{C}_{(0)}\right), \mu \right)$ and $\mathscr {F}_{t}: = \sigma \left(B_{s}, s \leqslant t\right), \mathscr {F}_{\infty } = \sigma \left(\bigcup_{t \geqslant 0} \mathscr {F}_{t}\right)$, the filtration generated by $B$. Show that for $F \in \mathscr {F}_{\infty }$ the following assertions are equivalent: