Posted on

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 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

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

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

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|偏微分方程代考Partial Differential Equations代写|Operator-valued angle bracket process

Let $M, N \in \mathcal{M}^2(U)$. Denote by $\langle\mathrm{M}, \mathrm{N}\rangle$ the unique predictable process, with trajectories having a locally bounded variation, for which
$$\langle M(t), N(t)\rangle_U-\langle M, N\rangle_t, \quad t \geq 0,$$
is a martingale. By the Doob-Meyer decomposition, the process $\langle M, N\rangle$ always exists (see Remark 3.46) and is called the angle bracket.

In this section we introduce the so-called operator angle bracket $\langle\langle M, N\rangle\rangle$, and in Theorem $8.2$ we will show the absolute continuity of the operator angle bracket with respect to the angle bracket. The relevant density is called the martingale covariance. It plays an important role in the construction of the stochastic integral with respect to $M$. In particular it appears in the fundamental isometric formula.
Denote by $L_1(U)$ the space of all nuclear operators on $U$ equipped with the nuclear norm; see Appendix A. Then $L_1(U)$ is a separable Banach space. Recall that, given $x, y, z \in U, x \otimes y(z)=\langle y, z\rangle_U x$. It is easy to show that $x \otimes y \in$ $L^1(U)$ and $|x \otimes y|_{L_1(U)}=|x|_U|y|_U$. We denote by $L_1^{+}(U)$ the subspace of $L_1(U)$ consisting of all self-adjoint non-negative nuclear operators. If $M \in \mathcal{M}^2(U)$ then the process $(M(t) \otimes M(t), t \geq 0)$ is an $L_1(U)$-valued right-continuous process such that
$$\mathbb{E}|M(t) \otimes M(t)|_{L_1(U)}=\mathbb{E}|M(t)|_U^2 \leq \mathbb{E}|M(T)|_U^2<\infty, \quad t \geq 0 .$$
We will need the following result.

## 数学代写|偏微分方程代考Partial Differential Equations代写|Construction of the stochastic integral

To deal with stochastic equations one needs the concept of the stochastic integral, $I_t^M(\Psi):=\int_0^t \Psi(s) \mathrm{d} M(s)$, where $M \in \mathcal{M}^2(U)$ and $\Psi(s, \omega)$ are operators from $U$ to another Hilbert space $H$. As for real-valued martingales, first we define the stochastic integral for simple processes $\Psi$. Then, in the next section, we extend the class of integrands using the isometric formula (8.3) below. The isometric formula in the general case appeared for the first time in Métivier and Pistone (1975). We will denote by $Q$ the martingale covariance of $M$ introduced in Definition 8.3.
Definition 8.5 Let $L(U, H)$ be the Banach space of continuous linear operators from $U$ into $H$. An $L(U, H)$-valued stochastic process $\Psi$ is said to be simple if there exist a sequence of non-negative numbers $t_0=0<t_1<\cdots<t_m$, a sequence of operators $\Psi_j \in L(U, H), j=1, \ldots, m$, and a sequence of events $A_j \in \mathcal{F}{t_j}, j=$ $0, \ldots, m-1$, such that $$\Psi(s)=\sum{j=0}^{m-1} \chi_{A_j} \chi_{\left(t_j, t_{j+1}\right]}(s) \Psi_j, \quad s \geq 0 .$$
We shall denote by $\mathcal{S}:=\mathcal{S}(U, H)$ the class of all simple processes with values in $L(U, H)$. For a simple process $\Psi$, we set
$$I_t^M(\Psi):=\sum_{j=0}^{m-1} \chi_{A_j} \Psi_j\left(M\left(t_{j+1} \wedge t\right)-M\left(t_j \wedge t\right)\right), \quad t \geq 0$$
Let $L_{(H S)}(U, H)$ be the space of all Hilbert-Schmidt operators from $U$ into $H$ equipped with the Hilbert-Schmidt norm $|\cdot|_{L_{(H S)}(U, H)}$. We prove the isometric formula first for simple processes.

# 偏微分方程代写

## 数学代写|偏微分方程代考Partial Differential Equations代写|Operator-valued angle bracket process

$$\langle M(t), N(t)\rangle_U-\langle M, N\rangle_t, \quad t \geq 0,$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|Construction of the stochastic integral

4. 这个结果可以推广到更高的维度。。

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。