Applications of Itô’s formula

State and prove the $d$ - dimensional version of Lemma 19.1.

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.

Let $(B(t))_{t \geqslant 0}$ be a $B M^{1}$ and $T < \infty$. Define $\beta_{T} = \exp \left(\xi B(T) - \frac{1}{2} \xi^{2} T\right), \mathbb {Q}: = \beta_{T} \mathbb {P}$ and $W(t): = B(t) - \xi t, t \in [0, T]$. Verify by a direct calculation that for all $0 < t_{1} < \cdots < t_{n}$ and $A_{1}, \ldots , A_{n} \in \mathscr {B}(\mathbb {R})$

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}, \Phi (y): = \mathbb {P}\left(B_{1} \leqslant y\right)$, and set $X_{t}: = B_{t}+\alpha t$ for some $\alpha \in \mathbb {R}$. Use Girsanov's theorem to show for all $0 \leqslant x \leqslant y, t > 0$

Let $X_{t}$ be as in Problem 19.5 and set $\widehat{\tau }_{b}: = \inf \left\{ t \geqslant 0: X_{t} = b\right\}$.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$, denote by $\mathcal{H}_{T}^{2}$ the space used in Lemma 19.10 and pick $\xi_{1}, \ldots , \xi_{n} \in \mathbb {R}$ and $0 = t_{0} < t_{1} < \cdots < t_{n} = T$. Show that $\exp \left[i \sum_{j = 1}^{n} \xi_{j} B_{t_{j}}\right]$ is contained in $\mathcal{H}_{T}^{2} \oplus i \mathcal{H}_{T}^{2}$.

Let $(\mathcal{L},\langle \cdot , \cdot \rangle )$ be a Hilbert space and $\mathcal{H}$ some linear subspace. Show that $\mathcal{H}$ is a dense subset of $\mathcal{L}$ if, and only if, the following property holds: $\forall x \in \mathcal{L}: x \; \perp \! \! \! \perp \; \mathcal{H} \Longrightarrow x = 0$.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}, n \geqslant 1,0 \leqslant t_{1} < \cdots < t_{n} = T$ and let $Y$ an $\mathscr {F}_{T}$ measurable, integrable random variable. Assume that $\mathbb {E}\left[Y \mid \sigma \left(B_{t_{1}}, \ldots , B_{t_{n}}\right)\right] = 0$ for any choice of $n \geqslant 1$ and $0 \leqslant t_{1} < \ldots t_{n} = T$. Show that $Y = 0$ a.s.

Assume that $\left(B_{t}, \mathscr {G}_{t}\right)_{t \geqslant 0}$ and $\left(M_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ are independent martingales. Show that $\left(M_{t}\right)_{t \leqslant T}$ and $\left(B_{t}\right)_{t \leqslant T}$ are martingales for the enlarged filtration $\mathscr {H}_{t}: = \sigma \left(\mathscr {F}_{t}, \mathscr {G}_{t}\right)$.

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.

Let $f:[0, \infty ) \rightarrow \mathbb {R}$ be absolutely continuous. Show that $\int f^{a - 1} d f = f^{a} / a$ for all $a > 0$.

State and prove a $d$ - dimensional version of the Burkholder - Davis - Gundy inequalities (19.21) if $p \in [2, \infty )$.

We have seen in Lemma 19.27.a) that $\operatorname {supp}\left[d L_{t}^{0}(\omega )\right] \subset \left\{ t \geqslant 0: B_{t}(\omega ) = 0\right\}$ for almost all $\omega$. Show that $\operatorname {supp}\left[d L_{t}^{a}(\omega )\right] \subset \left\{ t \geqslant 0: B_{t}(\omega ) = a\right\}$ for all $\omega$ from a set $\Omega_{0}$ with measure 1 such that $\Omega_{0}$ does not depend on $a$.

The proof of Theorem 19.29 uses, implicitly, the following beautiful result due to Skorokhod [239] which is to be proved:

Lemma. Let $b:[0, \infty ) \rightarrow \mathbb {R}$ be a continuous function such that $b(0) \geqslant 0$. There exist unique functions $p, a:[0, \infty ) \rightarrow [0, \infty )$ such that

Show that in Lemma 19.32.a) the following stronger assertion holds: $\left(L_{t}\right)_{t \geqslant 0}$ is an additive functional, i.e. $L_{t+s} = L_{t}+L_{s} \circ \theta_{t}$ holds for all $s, t \geqslant 0$.

Use the strong Markov property to show the stationary and independent increments property in Lemma 19.32.c) directly, i.e. for all $0 = u_{0} < u_{1} < \cdots < u_{n}$ and $A_{1}, \ldots , A_{n} \in \mathscr {B}[0, \infty )$ we have

Let $\left(X_{t}, \mathscr {G}_{t}\right)$ be an adapted, real - valued process with right continuous paths and finite left limits. Assume that $\mathbb {P}\left(X_{t} - X_{s} \in A \mid \mathscr {G}_{S}\right) = \mathbb {P}\left(X_{t - s} \in A\right)$ holds for all $0 \leqslant s < t < \infty$ and $A \in \mathscr {B}(\mathbb {R})$. Show that $\left(X_{t}\right)_{t \geqslant 0}$ enjoys the independent and stationary increment properties (B1), (B2).