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# 数学代写|偏微分方程代考Partial Differential Equations代写|AMATH562 Comparing two integrals

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## 数学代写|偏微分方程代考Partial Differential Equations代写|Comparing two integrals

Assume that $p=2$. We will show that the stochastic integral with respect to a compensated Poisson measure, introduced above, can be regarded as a stochastic integral with respect to a square integrable martingale, described in Section 8.2. In fact, we will relate the integrand $X \in \mathcal{L}{\mu, T}^2$ to $\tilde{X} \in \mathcal{L}{\widehat{\pi}, T}^2$ in such a way that $I_t^{\widehat{\pi}}(X)=\int_0^t \tilde{X}(s) \mathrm{d} \widehat{\pi}(s)$.

For this purpose we regard $\widehat{\pi}(s, \cdot)$ as a $U$-valued random variable for a properly chosen Hilbert space $U$. Namely, we assume that $U$ is a Hilbert space such that the embedding of the RKHS space $\mathcal{H}=L^2(E, \mathcal{E}, \mu) \hookrightarrow U$ is Hilbert-Schmidt. Additionally we assume that $\mathcal{H}$ is dense in $U$. Then, under the identification of $\mathcal{H}$ with its dual space, $U^* \hookrightarrow \mathcal{H}=\mathcal{H}^* \hookrightarrow U$. By Proposition $7.9$, we identify $\widehat{\pi}(t)$ with the family $\left.\left(\langle\psi, \widehat{\pi}(t)\rangle, \psi \in U^\right)\right)$, where $\langle\cdot, \cdot\rangle$ is the duality on $U^ \times U$. In Section $7.3$ we started the construction by defining $\langle\psi, \widehat{\pi}(t)\rangle$ as the stochastic integral of the deterministic mapping. Thus, with the notation of Section 7.3,
$$\langle\psi, \widehat{\pi}(t)\rangle=\widehat{\pi}(t, \psi)=\int_0^t \int_E \psi(\xi) \widehat{\pi}(\mathrm{d} s, \mathrm{~d} \xi), \quad \psi \in \mathcal{H}$$
Since
$$I_t^{\widehat{\pi}}(\psi)=\int_0^t \int_E \psi(\xi) \widehat{\pi}(\mathrm{d} s, \mathrm{~d} \xi)$$
and, under the identification of $\psi$ with an $\left(\mathcal{H}^=\mathcal{H}\right)$-valued process, $\langle\psi, \widehat{\pi}(t)\rangle=$ $\int_0^t \psi \mathrm{d} \widehat{\pi}(s)$, it follows that, for a deterministic time-independent field $X, I_t^{\widehat{\pi}}(\psi)=$ $\int_0^t \tilde{\psi}(s) \mathrm{d} \widehat{\pi}(s)$, where $\tilde{\psi} \in \mathcal{H}^=L_{(H S)}(\mathcal{H}, \mathbb{R})$ is given by
$$\tilde{\psi}[\varphi]=\langle\psi, \varphi\rangle_{\mathcal{H}}=\int_E \psi(\xi) \varphi(\xi) \mu(\mathrm{d} \xi), \quad \varphi \in \mathcal{H}$$
Thus, for a simple field $X, I_t^{\widehat{\pi}}(X)=\int_0^t \tilde{X}(s) \mathrm{d} \widehat{\pi}(s)$, where $\tilde{X}$ is a simple process in $L_{(H S)}(\mathcal{H}, \mathbb{R})$ given by
$$\tilde{X}(s)[\varphi]=\int_E X(s)(\xi) \varphi(\xi) \mu(\mathrm{d} \xi), \quad \varphi \in \mathcal{H}, s \geq 0$$
By approximation arguments we obtain the following result.

## 数学代写|偏微分方程代考Partial Differential Equations代写|$L^p$-theory for vector-valued integrands

Assume that $M$ is a square integrable Lévy martingale in a Hilbert space $U$ with RKHS $\mathcal{H}$. So far, we have seen how to integrate processes with values in the space of linear, possibly unbounded, operators from $U$ or $\mathcal{H}$ into another Hilbert space $H$. A special role is played by the space of Hilbert-Schmidt operators. One may ask whether it is possible to develop a similar theory of stochastic integration in Banach spaces. Thus, given a Banach space $B$, we are looking for a subspace $\mathcal{R}$ of the space of linear operators from $U$ to $B$ such that, for a simple $\mathcal{R}$-valued process
$$\Psi=\sum_n \alpha_i \Psi_i \chi_{\left(t_i, t_{i+1}\right]},$$
where $\Psi_i \in \mathcal{R}$ and $\alpha_i$ is an $\mathcal{F}{t_1}$-measurable real-valued bounded random variable, we have $$\mathbb{E}\left|\int_0^T \Psi(s) \mathrm{d} M(s)\right|_B^q \leq C{T, q} \mathbb{E} \int_0^T|\Psi(s)|_{\mathcal{R}}^q \mathrm{~d} s, \quad T \geq 0,$$
for some positive $q$. This, however, requires some geometrical properties of $B$; see Brzeźniak (1997) and Neidhardt (1978).

# 偏微分方程代写

## 数学代写|偏微分方程代考Partial Differential Equations代写|Comparing two integrals

$$\langle\psi, \widehat{\pi}(t)\rangle=\widehat{\pi}(t, \psi)=\int_0^t \int_E \psi(\xi) \widehat{\pi}(\mathrm{d} s, \mathrm{~d} \xi), \quad \psi \in \mathcal{H}$$

$$I_t^{\hat{\pi}}(\psi)=\int_0^t \int_E \psi(\xi) \widehat{\pi}(\mathrm{d} s, \mathrm{~d} \xi)$$

$$\bar{\psi}[\varphi]=\langle\psi, \varphi\rangle_{\mathcal{H}}=\int_E \psi(\xi) \varphi(\xi) \mu(\mathrm{d} \xi), \quad \varphi \in \mathcal{H}$$

$$\bar{X}(s)[\varphi]=\int_E X(s)(\xi) \varphi(\xi) \mu(\mathrm{d} \xi), \quad \varphi \in \mathcal{H}, s \geq 0$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|$L^p$-theory for vector-valued integrands

$$\Psi=\sum_n \alpha_i \Psi_i \chi_{\left(t_i, t_{i+1}\right]},$$

$$\mathbb{E}\left|\int_0^T \Psi(s) \mathrm{d} M(s)\right|B^q \leq C T, q \mathbb{E} \int_0^T|\Psi(s)|{\mathcal{R}}^q \mathrm{~d} s, \quad T \geq 0,$$

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