Filtration Construction and the Usual Conditions

This exercise gives an example where $\left\{ \mathcal{F}_{t}^{00}\right\}$ defined by (1.1) is not right continuous. Let $\Omega =\left\{ a, b\right\}$, let $\mathcal{F}$ be the collection of all subsets of $\Omega$, and let $\mathbb {P}\left(\left\{ a\right\} \right)=\mathbb {P}\left(\left\{ b\right\} \right)=\frac{1}{2}$. Define

Calculate $\mathcal{F}_{t}^{00}=\sigma \left(X_{s} ; s \leq t\right)$ and show $\left\{ \mathcal{F}_{t}^{00}\right\}$ is not right continuous.

If $X$ is a stochastic process, let $\mathcal{F}_{t}^{00}, \mathcal{F}_{t}^{0}$, and $\mathcal{F}_{t}$ be defined by (1.1), (1.2), and (1.3), respectively. Show that $\left\{ \mathcal{F}_{t}\right\}$ is the minimal augmented filtration generated by $X$.

Let $\mathcal{F}_{t}^{00}$ and $\mathcal{F}_{t}^{0}$ be defined as in (1.1) and (1.2). Prove that if $X$ is $\mathcal{F}_{t}^{0}$ measurable, there exists $Z$ such that $Z$ is $\mathcal{F}_{t}^{00}$ measurable and $Y=Z$, a.s.

Let $\mathcal{F}_{t}^{00}$ and $\mathcal{F}_{t}^{0}$ be defined as in (1.1) and (1.2). The symmetric difference of two sets $A$ and $B$ is defined by $A \bigtriangleup B=(A \backslash B) \cup (B \backslash A)$. Prove that

Whether a filtration satisfies the usual conditions depends on the class of null sets and hence the probability measure involved matters. Suppose $\left\{ \mathcal{F}_{t}\right\}$ satisfies the usual conditions with respect to $\mathbb {P}, H$ is a bounded predictable process, $W$ a Brownian motion with respect to $\mathbb {P}, M$ defined by (13.3), and $\mathbb {Q}$ defined by (13.4). If $t_{0}>0$ and $A \in \sigma \left(W_{s} ; s \leq t_{0}\right)$, show $\mathbb {P}\left(A\right)=0$ if and only if $\mathbb {Q}(A)=0$.

Let $\left(\mathscr {F}_{t}\right)_{t \geqslant 0}$ be a nondecreasing flow of $\sigma$-algebras (not necessarily complete) in a complete probability space $(\Omega , \mathscr {F}, \mathbb {P})$. Then $\overline{\mathscr {F}}_{t+} = \overline{\mathscr {F}_{t+}}$, i.e. the operations of augmentation by the class of $\mathbb {P}$-null sets and of right-continuous closure of the filtration commute (see Exercise 2).

For a filtration $\left(\mathscr {F}_{t}\right)_{t \geqslant 0}$ set $\overline{\mathscr {F}}_{t} = \sigma \left\{ \mathscr {F}_{t}, \mathscr {N}\right\}$, where $\mathscr {N}$ is the class of $\mathbb {P}$-null sets, and $\overline{\mathscr {F}}_{t+} = \bigcap_{s > t} \overline{\mathscr {F}}_{s}$, $t \geqslant 0$.

Let $\left(X_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a submartingale (martingale) with a.s. right-continuous sample paths. Suppose that for each $t \in \mathbb {R}_{+}$ there exists $\delta = \delta (t) > 0$ such that

Show that then $\bar{X} = \left(X_{t}, \overline{\mathscr {F}}_{t+}\right)$ is a submartingale (martingale).

Verify that condition (76) holds for the Wiener and Poisson processes (recall that the sample paths of the Wiener process are a.s. continuous, and the sample paths of the Poisson process are a.s. right-continuous).

Let $(\Omega , \mathcal{F}, \mathbb {P})$ be a probability space. A sequence of nested sigma-algebras $\mathcal{F}_{1} \subset \mathcal{F}_{2} \subset \ldots \subset \mathcal{F}$ is called a flow or filtration. Is it true that the system of sets $\bigcup_{n=1}^{\infty } \mathcal{F}_{n}$ is a $\sigma$-algebra?

The flow of sigma-algebras defined in problem 8 models a flow of information. Suppose one person thinks of a random number from $[0,1]$, and a second person tries to guess it. At each moment of time, the second person, knowing that the number belongs to some interval, divides that interval in half, and the first person says in which of the intervals the chosen number lies. Thus, with each moment of time we learn more and more information about the random number. Describe the flow of sigma-algebras in this problem.

Let $\left(X_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a martingale and denote by $\mathscr {F}_{t}^{*}$ be the completion of $\mathscr {F}_{t}$ (completion means to add all subsets of $\mathbb {P}$ - null sets).

Let $\left(X_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a submartingale with continuous paths and $\mathscr {F}_{t+} = \bigcap_{u > t} \mathscr {F}_{u}$. Show that $\left(X_{t}, \mathscr {F}_{t+}\right)_{t \geqslant 0}$ is again a submartingale.

Assume that the processes in Lemma 5.7 or in Lemma 5.8 are only "almost surely right continuous", resp. "almost surely continuous". Identify all steps in the proofs where this becomes relevant and show that the assertions remain valid if the filtrations $\mathscr {F}_{t}^{X}$ and $\mathscr {F}_{t+}^{X}$ are completed, so that they contain all subsets of measurable null sets.