Continuous semimartingales

Let $W_{t}$ be a Brownian motion started at 1 and $T_{0}=\inf \left\{ t>0: W_{t}=0\right\}$. Is $M_{t}=W_{t \wedge T_{0}}$ a square integrable martingale? A locally square integrable martingale? A uniformly integrable martingale? A martingale? A local martingale? A semimartingale?

Prove that if $M$ is a submartingale such that the paths of $M$ are continuous, sup ${ }_{t}\left|M_{t}\right|$ is integrable, and $S \leq T$ are finite stopping times, then $\mathbb {E}\left[M_{T} \mid \mathcal{F}_{S}\right] \geq M_{S}$. Note that the last part of the proof of Proposition 9.3 breaks down here.

Suppose $M_{t}$ is a local martingale with continuous paths. Show that if $N>0, T_{N}=\inf \{ t$ : $\left.\left|M_{t}\right| \geq N\right\}$, and $M_{t}^{N}=M_{t \wedge T_{N}}$, then $M^{N}$ is a uniformly integrable martingale.

Suppose $W_{t}^{1}$ and $W_{t}^{2}$ are two independent Brownian motions, $t_{0}>0$, and $M_{t}^{i}=W_{t \wedge t_{0}}, i=1,2$. Show $\left\langle M^{1}, M^{2}\right\rangle_{t}=0$.

Show that the norm defined in (9.10) is complete.

Let $Z_{t}$ be a bounded supermartingale with continuous paths that is constant from some time $t_{0}$ on. Show that for each $t$

We mentioned that one can prove the existence of $\left\langle M\right\rangle$ without using the Doob-Meyer theorem. Here is how that argument starts. Let $M$ be a bounded continuous martingale and for each $n$, define

Here $[x]$ is the integer part of $x$. Prove that for each $t>0$, $\mathbb {E}\left[\left|I_{n}(t)-I_{m}(t)\right|^{2}\right] \rightarrow 0$ as $n, m \rightarrow \infty$. One can then define $\left\langle M\right\rangle_{t}$ as the $L^{2}$ limit of $I_{n}(t)$.

Hint: If $n>m$, note that