Chapter 10, Martingale Spaces $\mathcal{H}^1$ and $\mathcal{B M O}$ Video Solutions, Semimartingale Theory and Stochastic Calculus

Problem 1

Let $M$ be a local martingale and $H$ be an optional process such that $H . M \in \mathcal{M}^2$. Then for any stopping time $T$ we have
$$
E\left[\left((H . M)_{\infty}-(H . M)_T\right)^2 \mid \mathcal{F}_T\right] \leq \boldsymbol{E}\left[\int_{\mid T, \infty} H_s^2 d[M]_a \mid \mathcal{F}_T\right]
$$

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00:42

Problem 2

Let $M \in \mathcal{B M O}$ and $H$ be an optional process with $|H| \leq 1$. Then $\|H . M\|_{\text {BMO }} \leq \sqrt{5}\|M\|_{\text {BMO }}$.

Ankit Gupta

Numerade Educator

02:16

Problem 3

$10.3 \mathcal{H}^{1, x}=\mathcal{H}^1 \cap \mathcal{M}_{\mathrm{loc}}^{\mathrm{c}}$ and $\mathcal{H}^{1, d}=\mathcal{H}^1 \cap \mathcal{M}_{\mathrm{loc}}^{\mathrm{d}}$ are all closed subspaces of $\mathcal{H}^1$.

Uma Kumari

Numerade Educator

Problem 4

Let $M \in B M \mathcal{O}_0$. If $\Delta M \geq-1+\varepsilon, \varepsilon \in[0,1]$, then $\mathcal{E}(M) \in \mathcal{M}$.

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03:05

Problem 5

Let $M \in \mathcal{M}_{\mathrm{loc}, 0}$.
1) If $E\left[\exp \left\{\frac{1}{2}(M)_{\infty}\right\}\right]<\infty$, then $\mathcal{E}(M) \in \mathcal{M}$.
2) If $E\left[\exp \left\{\frac{r}{2}\langle M\rangle_{\infty}\right\}\right]<\infty, r>1$, then $\mathcal{E}(M) \in \mathcal{H}^p, p=\frac{r^2}{2 r-1}$.

Harmender Singh Yadav

Numerade Educator

Problem 6

Let $M \in \mathcal{M}_{\text {loc }}$ and for any stopping time $T, E\left[\left|\Delta M_T\right| I_{T<\infty}\right] \mid<$ $\infty$. Then
$$
[M \rightarrow]=\left[[M]_{\infty}<\infty\right] \text { a.s.. }
$$

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03:38

Problem 7

Let $A \in \mathcal{A}_{\mathrm{loc}}^{+}$with $\bar{A}_{\infty}=\infty$ and for any stopping time $T$ $E\left[\Delta A_T I_{T<\infty)}\right]<\infty$. Then
$$
\lim _{t \rightarrow \infty} \frac{A_t}{\bar{A}_t}=1, \quad \text { a.s... }
$$

Cinsy Krehbiel

Numerade Educator

03:05

Problem 8

Let $M \in \mathcal{M}_{\text {loe }}$. Let
$$
B=\left\langle M^c\right\rangle+\sum \frac{\Delta M^2}{1+|\Delta M|} .
$$
and $A$ be the compensator of $B$.
1) $\left[A_{\infty}<\infty\right] \subset[M \rightarrow]$ a.s..
2) If for any stopping time $T, E\left[\left|\Delta M_T\right| I_{[T<\infty]}\right]<\infty$,
$$
\left[A_{\infty}<\infty\right]=[M \rightarrow] \text { a.s.. }
$$
3) If $E\left[A_{\infty}\right]<\infty, M \in \mathcal{M}$.

Harmender Singh Yadav

Numerade Educator

Problem 9

Let $M$ be a martingale, and $\sup _{t \geq 0} E\left[\left|M_t\right|\right]<\infty$. Let $N \in \mathcal{M}_{\text {loc }}$, and $[N] \leq[M]$. Then
$$
P([N \rightarrow])=1 .
$$

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01:40

Problem 10

Let $X \in S$, and $p>1$. Set
$$
\|X\|_{S \nu}=\inf \left\{\left\|\sqrt{[M]_{\infty}}+\int_{[0, \infty \mid}\left|d A_s\right|\right\|_p: X=M+A_{+}\right.
$$

Carson Merrill

Numerade Educator

Problem 11

Let $p \geq 1, q \geq 1, r \geq 1$ and $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$. Suppose $X \in \mathcal{S}^p$ and $H$ is a predictable process with $\left\|H_{\infty}^*\right\|_Q<\infty$. Then $H$ is $X$-integrable, and
$$
\begin{aligned}
& \|H . X\|_{w_r} \leq\left\|H_{\infty}^*\right\|_0\|X\|_{S p}, \\
& \left\|(H . X)_{\infty}^*\right\|_r \leq C_r\left\|H_{\infty}^*\right\|_q\|X\|_{S p},
\end{aligned}
$$
where $C_{\mathrm{r}}$ is a constant, depending on $r$ only.

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06:50

Problem 12

Let $X, X^{(n)} \in \mathcal{S}^p$. We say that $\left(X^{(n)}\right)$ converges prelocally in $\mathcal{S}^p$ to $X$ if there exists a sequence $\left(T_k\right)$ of stopping times with $T_k \uparrow \propto$ such that for each $k$
$$
\lim _{n \rightarrow \infty}\left\|\left(X^{(n)}-X\right)^{T_k-}\right\| s p=0 .
$$

The following two statements are equivalent:
1) $\lim _{n \rightarrow \infty} \sum_{k=1}^{\infty} 2^{-k} E\left[\left(X^{(n)}-X\right)_k^* \wedge 1\right]=0$.
2) From each subsequence of $\left(X^{(n)}\right)$ one can extract a subsoquence, which converges prelocally in $\mathcal{S}^p$ to $X$ (see Problem 8.20).

Supratim Pal

Numerade Educator