Some applications of Itô's formula

Show that $W_{t}$ and $W_{t}^{2}-t$ are local martingales, where $W$ is defined in the statement of Theorem 12.2 .

Suppose $\left\{ \mathcal{F}_{t}\right\}$ is a filtration satisfying the usual conditions, $X$ is a Brownian motion with respect to $\left\{ \mathcal{F}_{t}\right\}$, and $T$ is a finite stopping time with respect to this same filtration. Let $Y$ be another Brownian motion that is independent of $\left\{ \mathcal{F}_{t}\right\}$ and define

Show that $Z$ is a Brownian motion (although not necessarily with respect to $\left\{ \mathcal{F}_{t}\right\}$ ).

Suppose $M_{t}$ is a continuous local martingale with respect to a filtration $\left\{ \mathcal{F}_{t}\right\}$ satisfying the usual conditions, $T$ is a stopping time with respect to $\left\{ \mathcal{F}_{t}\right\}$, and $\langle M\rangle_{t}=t \wedge T$. Prove that $M_{t \wedge T}$ has the same law as a Brownian motion stopped at time $T$.

Here is a multidimensional version of Lévy's theorem. Let $\left\{ \mathcal{F}_{t}\right\}$ be a filtration satisfying the usual conditions. Suppose $\left(M_{t}^{1}, \ldots , M_{t}^{d}\right)$ is a $d$-dimensional process such that each component $M_{t}^{i}$ is a continuous martingale with respect to $\left\{ \mathcal{F}_{t}\right\}$ with $\left\langle M^{i}\right\rangle_{t}=t$. Suppose that $\left\langle M^{i}, M^{j}\right\rangle_{t}=0$ if $i \neq j$. Prove that $\left(M_{t}^{1}, \ldots , M_{t}^{d}\right)$ is a $d$-dimensional Brownian motion.

Let $\left\{ \mathcal{F}_{t}\right\}$ be a filtration satisfying the usual conditions. Let $A_{t}$ be a strictly increasing continuous process adapted to $\left\{ \mathcal{F}_{t}\right\}$ with $\lim_{t \rightarrow \infty } A_{t}=\infty$, a.s. Suppose $\left(M_{t}^{1}, \ldots , M_{t}^{d}\right)$ is a $d$-dimensional process such that each component $M_{t}^{i}$ is a continuous martingale with respect to $\left\{ \mathcal{F}_{t}\right\}$ and $\left\langle M^{i}\right\rangle_{t}=A_{t}$. Suppose that $\left\langle M^{i}, M^{j}\right\rangle_{t}=0$ if $i \neq j$. Prove that $\left(M_{t}^{1}, \ldots , M_{t}^{d}\right)$ is a time change of $d$-dimensional Brownian motion.

Suppose $M$ is a continuous local martingale such that $\langle M\rangle_{t}$ is deterministic. Prove that $M$ is a Gaussian process.

Suppose $M$ is a continuous local martingale with $M_{0}=0$, a.s. Show that there exists a Brownian motion $W$, an increasing process $\tau_{t}$, and a stopping time $T$ such that $M_{t}=W_{\tau_{\iota } \wedge T}$ for all $t$.

Let $M_{t}$ be a continuous local martingale. Show that the events $\left(M_{\infty }^{*}<\infty \right)$ and $\left(\langle M\rangle_{\infty }<\infty \right)$ differ by at most a null set.

Let $M_{t}$ be a continuous local martingale. Prove that

Suppose $X$ and $Y$ are continuous semimartingales and $t_{0}>0$. Prove that

converges to $\langle X, Y\rangle_{t_{0}}$ in probability.

Let $p>0$. Suppose $X$ and $Y$ are non-negative random variables, $\beta >1, \delta \in (0,1)$, and $\varepsilon \in \left(0, \beta^{-p} / 2\right)$ such that

for all $\lambda >0$. This inequality is known as a good- $\lambda$ inequality. Prove that there exists a constant $c$ (depending on $\beta , \delta , \varepsilon$, and $p$ but not $X$ or $Y$ ) such that

Hint: First assume $X$ is bounded. Write

Multiply by $p \lambda^{p-1}$, integrate over $\lambda$, and use the fact that $\varepsilon <\beta^{-P} / 2$.

Use Exercise 12.11 to prove that the Burkholder-Davis-Gundy inequalities hold for all $p>0$.

Hint: Use time change to reduce to the case of a Brownian motion $W$. If $T$ is a stopping time and $U=\inf \left\{ t: W_{T}^{*}>\lambda \right\}$, write

Condition on $\mathcal{F}_{U}$, use Theorem 4.2, and notice that $\mathbb {P}\left(U<\infty \right)=\mathbb {P}\left(W_{T}^{*}>\lambda \right)$.

Define the $H^{1}$ norm of a martingale by

Prove that this is a norm. Does there exist a uniformly integrable continuous martingale that is not in $H^{1}$ ?

Let $W$ be a Brownian motion and let $T$ be a stopping time. Prove that if $\mathbb {E}\left[T^{1 / 2}\right]<\infty$, then $\mathbb {E}\left[W_{T}\right]=0$.

Suppose $W=\left(W^{1}, \ldots , W^{d}\right)$ is a $d$-dimensional Brownian motion started at 0 , and let $\left\{ \mathcal{F}_{t}\right\}$ be the minimal augmented filtration of $W$. Suppose $Y$ is a $\mathcal{F}_{1}$ measurable random variable with mean zero and finite variance. Prove there exist predictable processes $H^{1}, \ldots , H^{d}$ such that $\mathbb {E}\left[\int_{0}^{1}\left(H_{s}^{i}\right)^{2} d s\right]<\infty$ for each $i$ and

Suppose $W$ is a Brownian motion and $H$ is adapted, bounded, and right continuous. Let $t \geq 0$. Show

converges in probability to $H_{t}$.

Let $W$ be a Brownian motion and $\alpha >0$. Show that

is infinite almost surely if $\alpha \geq 1$ but finite almost surely if $\alpha <1$.

Here is a useful inequality. Suppose $A$ is an increasing process with $A_{0}=0$, a.s., and suppose there exists a non-negative random variable $B$ such that for each $t$,

Prove that for each integer $p \geq 1$, there exists a constant $c_{p}$ depending only on $p$ such that

Hint: Write

take expectations, and use Proposition 3.14.

Let $W$ be a one-dimensional Brownian motion with filtration $\left\{ \mathcal{F}_{t}\right\}$ and let $f(r, s)$ be a deterministic function. Define the multiple stochastic integral by

provided

and similarly for higher-order multiple stochastic integrals.