Time change of continuous martingales (Dambis-Dubins-Schwarz)

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

Let $\left(I_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a continuous adapted process with values in $[0, \infty )$ and a.s. increasing sample paths. Set for $u \geqslant 0$

Show that

Show the following addition to Lemma 19.16:

and $\tau (s) \geqslant t \Longleftrightarrow a(t - ) \leqslant s$.

Show that, in Theorem 19.17, the quadratic variation $\left\langle M\right\rangle_{t}$ is a $\mathscr {G}_{t}$ stopping time.