Lévy's characterization of Brownian motion

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

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.

Prove that the equation

where $\sigma (x) = \operatorname {sgn} x$, has at least two solutions (but "weak" ones). Moreover, on some probability spaces this equation may have no "strong" solution at all.

Let $\operatorname {sign}(x) = 1$ if $x > 0$ and $\operatorname {sign}(x) = -1$ otherwise.

Let $X$ and $Y$ be independent standard Wiener processes. Show that, with $R_{t}^{2} = X_{t}^{2}+Y_{t}^{2}$,

is a Wiener process. [Hint: Use Theorem (13.8.11) and Example (12.7.10).] Hence show that the squared process $R^{2}$, called the squared Bessel process, satisfies

Generalize this conclusion to $n$ dimensions.

Let $B_{t} = \left(b_{t}, \beta_{t}\right), t \geqslant 0$ be a $\mathrm{BM}^{2}$ and set $r_{t}: = \left|B_{t}\right| = \sqrt{b_{t}^{2}+\beta_{t}^{2}}$.

Let $B_{t} = \left(b_{t}, \beta_{t}\right), t \geqslant 0$ be a $\mathrm{BM}^{2}$ and $f(x+i y) = u(x, y)+i v(x, y), x, y \in \mathbb {R}$, an analytic function. If $u_{x}^{2}+u_{y}^{2} = 1$, then $\left(u\left(b_{t}, \beta_{t}\right), v\left(b_{t}, \beta_{t}\right)\right), t \geqslant 0$, is a $\mathrm{BM}^{2}$.

Show that $\beta_{t} = \int_{0}^{t} \operatorname {sgn}\left(B_{s}\right) d B_{s}$ is a $\mathrm{BM}^{1}$.

Let $\left(N_{t}\right)_{t \geqslant 0}$ be a Poisson process with intensity $\lambda = 1$ (see Problem 10.1 for the definition). Show that for $\mathscr {F}_{t}^{N}: = \sigma \left(N_{r}: r \leqslant t\right)$ both $X_{t}: = N_{t} - t$ and $X_{t}^{2} - t$ are martingales. Explain why this does not contradict Theorem 19.5.

Let $B_{t} = \left(b_{t}, \beta_{t}\right), t \geqslant 0$, be a two - dimensional Brownian motion, i.e. $\left(b_{t}\right)_{t \geqslant 0},\left(\beta_{t}\right)_{t \geqslant 0}$ are independent one - dimensional Brownian motions. Show that for all $\sigma_{1}, \sigma_{2} > 0$

is a one - dimensional Brownian motion.