Martingale representation theorem

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 $P_{t}$ is a Poisson process with parameter $\lambda ,\left\{ \mathcal{F}_{t}\right\}$ is the minimal augmented filtration for $P$, and $M_{t}=P_{t}-\lambda t$. Suppose $Y$ is a $\mathcal{F}_{1}$ measurable random variable with finite mean and variance. Prove that there exists a predictable process $H$ such that

Let $P_{1}$ and $P^{2}$ be two independent Poisson processes with the same parameter. Let $X_{t}=P_{t}^{1}-P_{t}^{2}$ and let $\left\{ \mathcal{F}_{t}\right\}$ be the minimal augmented filtration for $X$. Find a bounded mean zero random variable $Y$ that is $\mathcal{F}_{1}$ measurable which does not satisfy

for any predictable process $H$.

Suppose $W$ is a Brownian motion and $\left\{ \mathcal{F}_t\right\}$ is its minimal augmented filtration. Let $T$ be a bounded stopping time with respect to $\left\{ \mathcal{F}_t\right\}$. Suppose $Y$ is a $\mathcal{F}_T$ measurable random variable with $\mathbb {E}\left[Y^2\right]<\infty$. Show that there exists a predictable process $H_s$ with $\mathbb {E}\left[\int_0^T H_s^2 d s\right]<\infty$ such that $Y=\mathbb {E}\left[Y\right]+\int_0^T H_s d W_s$, a.s.

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 $\left(B_{t}\right)_{t \geqslant 0}$ 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.