The general theory of processes

Show that if $S_{1}, \ldots , S_{n}$ are predictable stopping times, then so are $S_{1} \wedge \cdots \wedge S_{n}$ and $S_{1} \vee \cdots \vee S_{n}$.

If $A_{t}$ is a predictable process with paths that are right continuous with left limits and $a>0$, show $T=\inf \left\{ t>0: \Delta A_{t}>a\right\}$ is a predictable stopping time.

Show that if $T$ is a predictable stopping time, $B \in \mathcal{F}_{T_{-}}$, and $T_{B}(\omega )$ is defined to be equal to $T(\omega )$ if $\omega \in B$ and equal to $\infty$ otherwise, then $T_{B}$ is a predictable stopping time.

Let $X$ be a bounded adapted right-continuous process, let $\varepsilon >0$, let $U_{0}=0$, a.s., and define $U_{i}$ by (16.3) for $i \geq 1$. Show each $U_{i}$ is a stopping time.

Let $A$ be a predictable increasing process and let $S_{k}$ be the $k$ th time $A$ jumps more than $\varepsilon$. Thus $S_{0}=0$, a.s., and $S_{k+1}=\inf \left\{ t>S_{k}: \Delta A_{t}>\varepsilon \right\}$. Show each $S_{k}$ is a predictable stopping time.

Show that the stopping times $T_{n}$ defined in the proof of Proposition 16.23 are predictable.

From that proof: $R_{1}, R_{2}, \ldots$ is a countable collection of predictable stopping times (constructed from a decomposition of certain other predictable stopping times $U_{m}^{i}$). Define $T_{1}=R_{1}$; define $T_{2}$ by setting $T_{2}(\omega )=R_{2}(\omega )$ if $R_{2}(\omega ) \neq R_{1}(\omega )$ and $T_2(\omega ) = \infty$ otherwise; and in general set $T_{n}(\omega )=R_{n}(\omega )$ if $R_{n}(\omega ) \neq R_{1}(\omega ), \ldots , R_{n-1}(\omega )$ and $T_{n}(\omega )=\infty$ otherwise.

Show that if $P_{t}$ is a Poisson process, then $\left({ }^{p} P\right)_{t}=P_{t-}$.

Show that if $X$ is bounded and measurable with respect to the product $\sigma$-field $\mathcal{B}[0, \infty ) \times \mathcal{F}_{\infty }$, then ${ }^{p}\left({ }^{o} X\right)={ }^{p} X$.

Suppose $T$ is a totally inaccessible stopping time. Show that if $X=\; \mathbb {1}_{[T, T]}$, then ${ }^{p} X=0$.

If $P$ is a Poisson process with parameter $\lambda$, determine $P_{t}^{o}$ and $P_{t}^{p}$.

Show that $\mu_{o}$ defined in (16.11) is a measure.

Let $X_{t}$ be a continuous process and suppose there exists $K>0$ such that for all $t$,

Let $X_{\infty }^{*}=\sup_{t \geq 0}\left|X_{t}\right|$. Prove that there exists $a$ depending only on $K$ such that

A martingale $M$ is said to be in the space $BMO$ if

Let $M_{t}^{*}=\sup_{s \leq t}\left|M_{s}\right|$. Show that if $M$ is in $BMO$, then there exists $a>0$ such that

The name $BMO$ comes from the "bounded mean oscillation" spaces of harmonic analysis.