Martingales

If $W$ is a Brownian motion, show that

is a martingale.

Suppose $\left\{ \mathcal{F}_{t}\right\}$ is a filtration satisfying the usual conditions. Show that if $M_{t}$ is a submartingale and $\mathbb {E}\left[M_{t}\right]=\mathbb {E}\left[M_{0}\right]$ for all $t$, then $M$ is a martingale.

Let $X$ be a submartingale. Show that $\sup_{t \geq 0} \mathbb {E}\left[\left|X_{t}\right|\right]<\infty$ if and only if $\sup_{t \geq 0} \mathbb {E}\left[X_{t}^{+}\right]<\infty$.

Prove all parts of Proposition 3.8.

Proposition 3.8: Suppose $\left\{ \mathcal{F}_{t}\right\}$ satisfies the usual conditions. Then

1. $T$ is a stopping time if and only if $(T \leq t) \in \mathcal{F}_{t}$ for all $t$.

2. If $T=t$, a.s., then $T$ is a stopping time.

3. If $S$ and $T$ are stopping times, then so are $S \vee T$ and $S \wedge T$.

4. If $T_{n}, n=1,2, \ldots$, are stopping times with $T_{1} \leq T_{2} \leq \cdots$, then so is $\sup_{n} T_{n}$.

5. If $T_{n}, n=1,2, \ldots$, are stopping times with $T_{1} \geq T_{2} \geq \cdots$, then so is $\inf_{n} T_{n}$.

6. If $s \geq 0$ and $S$ is a stopping time, then so is $S+s$.

If $T_{n}$ is defined by (3.2), show $T_{n}$ is a stopping time for each $n$ and $T_{n} \downarrow T$.

This exercise gives an alternate definition of $\mathcal{F}_{T}$ which is more appealing, but not as useful. Suppose that $\left\{ \mathcal{F}_{t}\right\}$ satisfies the usual conditions. Show that $\mathcal{F}_{T}$ is equal to the $\sigma$-field generated by the collection of random variables $Y_{T}$ such that $Y$ is a bounded process with paths that are right continuous with left limits and $Y$ is adapted to the filtration $\left\{ \mathcal{F}_{t}\right\}$.

Suppose $\left\{ \mathcal{F}_{t}\right\}$ is a filtration satisfying the usual conditions. Show that if $T$ is a stopping time, then $T$ is $\mathcal{F}_{T}$ measurable.

Suppose $\left\{ \mathcal{F}_{t}\right\}$ is a filtration satisfying the usual conditions and $T$ is a stopping time. Show that if $S$ is a $\mathcal{F}_{T}$ measurable random variable with $S \geq T$, then $S$ is a stopping time.

This exercise demonstrates that the conclusion of Corollary 3.13 cannot be extended to submartingales. Find a filtration $\left\{ \mathcal{F}_{t}\right\}$ satisfying the usual conditions and a submartingale $X$ with respect to $\left\{ \mathcal{F}_{t}\right\}$ such that $X$ does not have a version with paths that are right continuous with left limits.

Suppose $\left\{ \mathcal{F}_{t}\right\}$ is a filtration satisfying the usual conditions. Show that if $S$ and $T$ are stopping times and $X$ is a bounded $\mathcal{F}_{\infty }$ measurable random variable, then

Hint: Let $Y_{t}=\mathbb {E}\left[X \mid \mathcal{F}_{t}\right]$ and $Z_{t}=Y_{t \wedge S}$. Show the left-hand side is equal to $Y_{S \wedge T}$.

A martingale or submartingale $M_{t}$ is uniformly integrable if the family $\left\{ M_{t}: t \geq 0\right\}$ is a uniformly integrable family of random variables. Show that if $M_{t}$ is a uniformly integrable martingale with paths that are right continuous with left limits, then $\left\{ M_{T} ; T\right.$ a finite stopping time $\}$ is a uniformly integrable family of random variables. Show this also holds if $M_{t}$ is a non-negative submartingale with paths that are right continuous with left limits.

This exercise weakens the conditions on the optional stopping theorem. Show that if $M_{t}$ is a uniformly integrable martingale that is right continuous with left limits and $T$ is a finite stopping time, then $\mathbb {E}\left[M_{T}\right]=\mathbb {E}\left[M_{0}\right]$.

Let $W$ be a Brownian motion and let $T$ be a stopping time with $\mathbb {E}\left[T\right]<\infty$. Prove that $\mathbb {E}\left[W_{T}\right]=0$ and $\mathbb {E}\left[W_{T}^{2}\right]=\mathbb {E}\left[T\right]$. This is not an easy application of the optional stopping theorem because we do not know that $W_{t \wedge T}$ is necessarily a uniformly integrable martingale.

Suppose that $\left(W_{t}^{1}, \ldots , W_{t}^{d}\right)$ is a $d$-dimensional Brownian motion. Show that if $i \neq j$, then $W_{t}^{i} W_{t}^{j}$ is a martingale.

Let $W_{t}$ be a Brownian motion, $b>0$, and $T=\inf \left\{ t>0: W_{t}=b\right\}$. Show $T<\infty$, a.s. Show $\mathbb {E}\left[T\right]=\infty$.

Hint: Take a limit in (3.11).

Suppose $W$ is a Brownian motion and $b>0$. If $S=\inf \left\{ t>0: W_{t}=b\right\}$, show that the Laplace transform of the density of $S$ is given by

Let $W_{t}$ be a Brownian motion. Show that if $\alpha >1 / 2$, then

Hint: Let $\alpha_{0} \in (1 / 2, \alpha )$, estimate

using (3.9), and then use the Borel-Cantelli lemma.

Let $W_{t}$ be a one-dimensional Brownian motion and $\alpha \in (0,1 / 2]$. Prove that

If $W$ is a Brownian motion and $b$ is a constant, then the process $X_{t}=W_{t}+b t$ is a Brownian motion with drift. Prove that if $b>0$, then