## 统计代写|随机过程代写stochastic process代考|STAT7004

statistics-lab™ 为您的留学生涯保驾护航 在代写随机过程stochastic process方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机过程stochastic process代写方面经验极为丰富，各种代写随机过程stochastic process相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|随机过程代写stochastic process代考|Global Carleman Estimate for Stochastic Parabolic

In this subsection, as a preliminary to prove Theorem 9.28 , we shall derive a global Carleman estimate for the following stochastic parabolic equation:
$$\begin{cases}d h-\sum_{j, k=1}^n\left(a^{j k} h_{x_j}\right){x_k} d t=f d t+g d W(t), & \text { in } Q, \ h=0, & \text { on } \Sigma, \ h(0)=h_0, & \text { in } G,\end{cases}$$ where $h_0 \in L{\mathcal{F}_0}^2\left(\Omega ; L^2(G)\right)$, while $f$ and $g$ are suitable stochastic processes to be given later.

We begin with the following known technical result (See [117, p. 4, Lemma 1.1] and [337, Lemma 2.1] for its proof), which shows the existence of a nonnegative function with an arbitrarily given critical point location in $G$.
Lemma 9.29. For any nonempty open subset $G_1$ of $G$, there is a $\psi \in C^{\infty}(\bar{G})$ such that $\psi>0$ in $G, \psi=0$ on $\Gamma$, and $|\nabla \psi(x)|>0$ for all $x \in \overline{G \backslash G_1}$.
In the rest of this section, we choose $\theta$ and $\ell$ as that in (9.88), and $\psi$ given by Lemma 9.29 with $G_1$ being any fixed nonempty open subset of $G$ such that $\overline{G_1} \subset G_0$. The desired Carleman estimate for $(9.94)$ is stated as follows:

Theorem 9.30. There is a constant $\mu_0=\mu_0\left(G, G_0,\left(a^{j k}\right){n \times n}, T\right)>0$ such that for all $\mu \geq \mu_0$, one can find two constants $\mathcal{C}=\mathcal{C}(\mu)>0$ and $\lambda_0=\lambda_0(\mu)>$ 0 such that for any $\lambda \geq \lambda_0, h_0 \in L{\mathcal{F}0}^2\left(\Omega ; L^2(G)\right), f \in L{\mathbb{F}}^2\left(0, T ; L^2(G)\right)$ and $g \in$ $L_{\mathbb{F}}^2\left(0, T ; H^1(G)\right)$, the corresponding solution $h \in L_{\mathbb{F}}^2\left(\Omega ; C\left([0, T] ; L^2(G)\right)\right) \cap$ $L_F^2\left(0, T ; H_0^1(G)\right)$ to $(9.94)$ satisfies
\begin{aligned} & \lambda^3 \mu^4 \mathbb{E} \int_Q \theta^2 \varphi^3 h^2 d x d t+\lambda \mu^2 \mathbb{E} \int_Q \theta^2 \varphi|\nabla h|^2 d x d t \ & \leq \mathcal{C} \mathbb{E}\left[\int_Q \theta^2\left(f^2+|\nabla g|^2+\lambda^2 \mu^2 \varphi^2 g^2\right) d x d t+\lambda^3 \mu^4 \int_{Q_0} \theta^2 \varphi^3 h^2 d x d t\right] \end{aligned}
Proof: We barrow some idea from [117]. We shall use Theorem 9.27 with $b^{j k}$ being replaced by $a^{j k}$ (and hence $\mathbf{u}=h$ ). The proof is divided into three steps.

## 统计代写|随机过程代写stochastic process代考|Global Carleman Estimate for Stochastic Parabolic

In this subsection, we derive an improved global Carleman estimate for the forward stochastic parabolic equation (9.94).

Throughout this subsection, $\mu=\mu_0$ and $\lambda \geq \lambda_0$ are given as that in Theorem 9.30, and $\theta$ and $\varphi$ are the same as that in the last subsection.

For any fixed $f, g \in L_F^2\left(0, T ; L^2(G)\right)$ and $h_0 \in L_{\mathcal{F}0}^2\left(\Omega ; L^2(G)\right)$, let $h$ denote the corresponding solution to the equation (9.94). Based on the Carleman estimate in Corollary 9.32 , we give below a “partial” null controllability result for the following controlled backward stochastic parabolic equation: $$\begin{cases}d r+\sum{j, k=1}^n\left(a^{j k} r_{x_j}\right){x_k} d t=\left(\lambda^3 \theta^2 \varphi^3 h+\chi{G_1} u\right) d t+R d W(t) & \text { in } Q, \ r=0 & \text { on } \Sigma, \ r(T)=0 & \text { in } G,\end{cases}$$
where $u \in L_{\mathbb{F}}^2\left(0, T ; L^2\left(G_1\right)\right)$ is the control variable and $(r, R)$ is the state variable.

Proposition 9.33. There exists a control $\hat{u} \in L_{\mathrm{F}}^2\left(0, T ; L^2\left(G_1\right)\right)$ such that the corresponding solution $(\hat{r}, \widehat{R}) \in\left(L_{\mathbb{F}}^2\left(\Omega ; C\left([0, T] ; L^2(G)\right)\right) \cap L_{\mathbb{F}}^2\left(0, T ; H_0^1(G)\right)\right)$ $\times L_{\mathbb{F}}^2\left(0, T ; L^2(G)\right)$ to $(9.108)$ with $u=\hat{u}$ satisfies $\hat{r}(0)=0$ in $G$, a.s. Moreover,
$$\mathbb{E} \int_Q \theta^{-2}\left(\hat{r}^2+\lambda^{-3} \varphi^{-3} \hat{u}^2+\lambda^{-2} \varphi^{-2} \widehat{R}^2\right) d x d t \leq \mathcal{C} \lambda^3 \mathbb{E} \int_Q \theta^2 \varphi^3 h^2 d x d t .$$

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|Global Carleman Estimate for Stochastic Parabolic

$$\left{d h-\sum_{j, k=1}^n\left(a^{j k} h_{x_j}\right) x_k d t=f d t+g d W(t), \quad \text { in } Q, h=0, \quad \text { on } \Sigma, h(0)=h_0,\right.$$

$$\lambda^3 \mu^4 \mathbb{E} \int_Q \theta^2 \varphi^3 h^2 d x d t+\lambda \mu^2 \mathbb{E} \int_Q \theta^2 \varphi|\nabla h|^2 d x d t \quad \leq \mathcal{C} \mathbb{E}\left[\int _ { Q } \theta ^ { 2 } \left(f^2+|\nabla g|^2+\lambda^2\right.\right.$$

## 统计代写|随机过程代写stochastic process代考|Global Carleman Estimate for Stochastic Parabolic

$$\left{d r+\sum j, k=1^n\left(a^{j k} r_{x_j}\right) x_k d t=\left(\lambda^3 \theta^2 \varphi^3 h+\chi G_1 u\right) d t+R d W(t) \quad \text { in } Q, r=0\right.$$

$$\mathbb{E} \int_Q \theta^{-2}\left(\hat{r}^2+\lambda^{-3} \varphi^{-3} \hat{u}^2+\lambda^{-2} \varphi^{-2} \widehat{R}^2\right) d x d t \leq \mathcal{C} \lambda^3 \mathbb{E} \int_Q \theta^2 \varphi^3 h^2 d x d t$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|随机过程代写stochastic process代考|STAT3061

statistics-lab™ 为您的留学生涯保驾护航 在代写随机过程stochastic process方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机过程stochastic process代写方面经验极为丰富，各种代写随机过程stochastic process相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|随机过程代写stochastic process代考|Proof of the Negative Null Controllability Result

In this subsection, we show the lack of null controllability for the system (9.58), presented in Theorem 9.19.

Proof of Theorem 9.19: First, we prove that the system (9.58) is not null controllable if $a_5(\cdot)=0$ in $(0, T) \times \Omega$, a.e.

Without loss of generality, we may assume that the coefficient $a_6(\cdot)$ in the system (9.58) is equal to 0 (Otherwise, we introduce a simple transformation $\tilde{y}=y, \tilde{z}(t)=e^{-\int_0^t a_6(s) d s} z(t)$ and $\tilde{u}=u$ and consider the system for the new state variable $(\tilde{y}, \tilde{z})$ and the control variable $\tilde{u})$. Then, by the system (9.58), and noting that $a_5(\cdot)=a_6(\cdot)=0$ in $(0, T) \times \Omega$, a.e., we find that $(\mathbb{E} y, \mathbb{E} z)$ solves
$$\begin{cases}(\mathbb{E} y)t-\sum{j, k=1}^n\left(a^{j k}(\mathbb{E} y){x_j}\right){x_k}=\mathbb{E}\left(a_1(t) y+a_2(t) z+\chi_{G_0} u\right) & \text { in } Q, \ (\mathbb{E} z)t-\sum{j, k=1}^n\left(a^{i j}(\mathbb{E} z){x_j}\right){x_k}=0 & \text { in } Q, \ \mathbb{E} y=\mathbb{E} z=0 & \text { on } \Sigma, \ (\mathbb{E} y)(0)=y_0,(\mathbb{E} z)(0)=z_0 & \text { in } G .\end{cases}$$
Since there is no control in the second equation of (9.73), $\mathbb{E} z$ cannot be driven to the rest for any time $T$ if $z_0 \neq 0$ in $G$.

Next, we prove that the system (9.58) is not null controllable if $a_8(\cdot) \neq 0$ in $(0, T) \times \Omega$, a.e. and $\frac{a_5(\cdot)}{a_8(\cdot)} \in L_{\mathrm{F}}^{\infty}(0, T)$.

In the following, we construct a nontrivial solution $(\alpha, \beta, K, R)$ to (9.60) such that $a_5(t) \alpha+a_8(t) K=0$ and $(\bar{\beta}, R)=0$ in $\bar{Q}$, a.s. For this purpose, we consider the following linear stochastic differential equation:
$$\left{\begin{array}{l} d \zeta-\lambda_1 \zeta d t=a_6(t) \zeta d t-\frac{a_5(t)}{a_8(t)} \zeta d W(t) \text { in }(0, T) \ \zeta(0)=1 \end{array}\right.$$

## 统计代写|随机过程代写stochastic process代考|Observability Estimate for Stochastic Parabolic

In this section, we shall establish an observability estimate for the stochastic parabolic equation (9.16). The main content of this section is taken from $[218,315]$
We have the following result.
Theorem 9.28. Let the condition (9.17) be satisfied. Then, solutions to (9.16) satisfy that, for any $z_0 \in L_{\mathcal{F}0}^2\left(\Omega ; L^2\left(G ; \mathbb{R}^m\right)\right)$, $$|z(T)|{L_{\mathcal{F}T}^2\left(\Omega ; L^2\left(G ; \mathbb{R}^m\right)\right)} \leq \mathcal{C} e^{\mathcal{C} R_2^2}|z|{L_F^2\left(0, T ; L^2\left(G_0 ; \mathbb{R}^m\right)\right)},$$
where $R_2$ is given in Corollary 9.5.
We shall give a proof of Theorem 9.28 in Subsection 9.5.3. For simplicity, we consider only the case $m=1$.

In this subsection, as a preliminary to prove Theorem 9.28 , we shall derive a global Carleman estimate for the following stochastic parabolic equation:
$$\begin{cases}d h-\sum_{j, k=1}^n\left(a^{j k} h_{x_j}\right){x_k} d t=f d t+g d W(t), & \text { in } Q \ h=0, & \text { on } \Sigma, \ h(0)=h_0, & \text { in } G\end{cases}$$ where $h_0 \in L{\mathcal{F}_0}^2\left(\Omega ; L^2(G)\right)$, while $f$ and $g$ are suitable stochastic processes to be given later.

We begin with the following known technical result (See [117, p. 4, Lemma 1.1] and [337, Lemma 2.1] for its proof), which shows the existence of a nonnegative function with an arbitrarily given critical point location in $G$.
Lemma 9.29. For any nonempty open subset $G_1$ of $G$, there is a $\psi \in C^{\infty}(\bar{G})$ such that $\psi>0$ in $G, \psi=0$ on $\Gamma$, and $|\nabla \psi(x)|>0$ for all $x \in \overline{G \backslash G_1}$.

In the rest of this section, we choose $\theta$ and $\ell$ as that in (9.88), and $\psi$ given by Lemma 9.29 with $G_1$ being any fixed nonempty open subset of $G$ such that $\overline{G_1} \subset G_0$. The desired Carleman estimate for $(9.94)$ is stated as follows:

Theorem 9.30. There is a constant $\mu_0=\mu_0\left(G, G_0,\left(a^{j k}\right){n \times n}, T\right)>0$ such that for all $\mu \geq \mu_0$, one can find two constants $\mathcal{C}=\mathcal{C}(\mu)>0$ and $\lambda_0=\lambda_0(\mu)>$ 0 such that for any $\lambda \geq \lambda_0, h_0 \in L{\mathcal{F}0}^2\left(\Omega ; L^2(G)\right), f \in L{\mathrm{F}}^2\left(0, T ; L^2(G)\right)$ and $g \in$ $L_{\mathbb{F}}^2\left(0, T ; H^1(G)\right)$, the corresponding solution $h \in L_{\mathbb{F}}^2\left(\Omega ; C\left([0, T] ; L^2(G)\right)\right) \cap$ $L_{\mathrm{F}}^2\left(0, T ; H_0^1(G)\right)$ to $(9.94)$ satisfies
\begin{aligned} & \lambda^3 \mu^4 \mathbb{E} \int_Q \theta^2 \varphi^3 h^2 d x d t+\lambda \mu^2 \mathbb{E} \int_Q \theta^2 \varphi|\nabla h|^2 d x d t \ & \leq \mathcal{C} \mathbb{E}\left[\int_Q \theta^2\left(f^2+|\nabla g|^2+\lambda^2 \mu^2 \varphi^2 g^2\right) d x d t+\lambda^3 \mu^4 \int_{Q_0} \theta^2 \varphi^3 h^2 d x d t\right] \end{aligned}

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|Proof of the Negative Null Controllability Result

$$\left{(\mathbb{E} y) t-\sum j, k=1^n\left(a^{j k}(\mathbb{E} y) x_j\right) x_k=\mathbb{E}\left(a_1(t) y+a_2(t) z+\chi_{G_0} u\right) \quad \text { in } Q,(\mathbb{E} z) t\right.$$

$\$ \$$Veft$$
d \zeta-\lambda_1 \zeta d t=a_6(t) \zeta d t-\frac{a_5(t)}{a_8(t)} \zeta d W(t) \text { in }(0, T) \zeta(0)=1
$$l正确的。 \ \$$

## 统计代写|随机过程代写stochastic process代考|Observability Estimate for Stochastic Parabolic

$$|z(T)| L_{\mathcal{F} T}^2\left(\Omega ; L^2\left(G ; \mathbb{R}^m\right)\right) \leq \mathcal{C} e^{\mathcal{C} R_2^2}|z| L_F^2\left(0, T ; L^2\left(G_0 ; \mathbb{R}^m\right)\right),$$

$\left{d h-\sum_{j, k=1}^n\left(a^{j k} h_{x_j}\right) x_k d t=f d t+g d W(t), \quad\right.$ in $Q h=0, \quad$ on $\Sigma, h(0)=h_0$,

$$\lambda^3 \mu^4 \mathbb{E} \int_Q \theta^2 \varphi^3 h^2 d x d t+\lambda \mu^2 \mathbb{E} \int_Q \theta^2 \varphi|\nabla h|^2 d x d t \quad \leq \mathcal{C} \mathbb{E}\left[\int _ { Q } \theta ^ { 2 } \left(f^2+|\nabla g|^2+\lambda^2\right.\right.$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|随机过程代写stochastic process代考|STAT3061

statistics-lab™ 为您的留学生涯保驾护航 在代写随机过程stochastic process方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机过程stochastic process代写方面经验极为丰富，各种代写随机过程stochastic process相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|随机过程代写stochastic process代考|Formulation of the Problems

The $(F, \Gamma)$-controllability concept introduced in Definition 5.6 (in Section 5.3 of Chapter 5) is quite general, which is very hard to be studied (even in the deterministic setting and in finite dimensions). In this chapter, we shall focus mainly on exact/null/approximate controllability problems for stochastic linear evolution equations in the abstract setting.

Throughout this chapter, $H, V$ and $U$ are three Hilbert spaces which are identified with their dual spaces, $\mathcal{L}2^0 \triangleq \mathcal{L}_2(V ; H)$, and $A$ is the generator of a $C_0$-semigroup ${S(t)}{t \geq 0}$ on $H$. In this chapter, $H$ and $U$ will serve as respectively the state and the control spaces for the systems under consideration unless other stated.

For any fixed $T>0$ and a control operator $B \in \mathcal{L}(U ; H)$, we begin with the following deterministic controlled evolution equation:
$$\left{\begin{array}{l} y_t(t)=A y(t)+B u(t) \quad \text { in }(0, T], \ y(0)=y_0, \end{array}\right.$$
where $y_0 \in H, y$ is the state variable, and $u\left(\in L^2(0, T ; U)\right)$ is the control variable. The exact/null/approximate controllability of (7.1) can be defined as that in Definition 5.6. For example, the system (7.1) is called exactly (resp. null) controllable at time $T$ if for any $y_0, y_T \in H$ (resp. $y_0 \in H$ ), one can find a control $u \in L^2(0, T ; U)$ so that the corresponding solution to (7.1) verifies $y(T)=y_T($ resp. $y(T)=0)$

## 统计代写|随机过程代写stochastic process代考|Well-Posedness of Stochastic Systems

In this section, we shall use the transposition method to establish the wellposedness of stochastic control systems (7.8) and (7.9).

In order to define transposition solutions to $(7.8)$, we introduce the following (backward stochastic) test equation evolved in $O$ :
$$\left{\begin{array}{l} d \xi(t)=-\left(A^* \xi(t)+\mathcal{F}(t)^* \xi(t)+\mathcal{G}(t)^* \Xi(t)\right) d t+\Xi(t) d W(t) \text { in }[0, \tau), \ \xi(\tau)=\xi_\tau . \end{array}\right.$$
In (7.14), $\tau \in[0, T]$ and $\xi_\tau \in L_{\mathcal{F}\tau}^2(\Omega ; O)$. Recall that we assume $\mathbb{F}$ is the natural filtration. Hence, by Theorem $4.10$, the equation $(7.14)$ admits a unique mild solution $(\xi, \Xi) \in L{\mathbb{F}}^2(\Omega ; C([0, \tau] ; O))$ $\times L_{\mathbb{F}}^2\left(0, \tau ; \mathcal{L}2(V ; O)\right)$ such that $$|(\xi, \Xi)|{L_F^2(\Omega ; C([0, \tau] ; O)) \times L_F^2\left(0, \tau ; \mathcal{L}2(V ; O)\right)} \leq \mathcal{C}(\mathcal{F}, \mathcal{G})\left|\xi\tau\right|{L{\mathcal{F}\gamma}^2(\Omega ; O)}$$ We make the following additional assumptions. Condition 7.1 There exists a sequence $\left{u_n\right}{n=1}^{\infty} \subset \mathcal{U}T$ such that $\mathcal{B} u_n \in$ $L{\mathbb{F}}^2\left(0, T ; O^{\prime}\right)$ and $\mathcal{D} u_n \in L_{\mathbb{F}}^2\left(0, T ; \mathcal{L}2\left(V ; O^{\prime}\right)\right)$ for each $n \in \mathbb{N}$ and $$\lim {n \rightarrow \infty} u_n=u \quad \text { in } \mathcal{U}T \text {. }$$ Condition 7.2 There exists a constant $\mathcal{C}>0$ such that for any $\tau \in[0, T]$ and $\xi\tau \in L_{\mathcal{F}\tau}^2(\Omega ; O)$, the solution $(\xi, \Xi)$ to (7.14) satisfies that $$\left|\mathcal{B}^* \xi+\mathcal{D}^* \Xi\right|{L_F^2(0, \tau ; U)} \leq \mathcal{C}\left|\xi_\tau\right|{L{\mathcal{F}\tau}^2(\Omega ; O)}$$ Remark 7.10. When $\mathcal{D}$ is bounded, i.e., $\mathcal{D} \in \mathcal{L}\left(U ; \mathcal{L}_2\left(V ; O^{\prime}\right)\right)$, the condition $\mathcal{D} u_n \in L{\mathbb{F}}^2\left(0, T ; \mathcal{L}2\left(V ; O^{\prime}\right)\right)$ in Condition $7.1$ holds automatically, and the inequality (7.17) in Condition $7.2$ is equivalent to the following estimate: $$\left|\mathcal{B}^* \xi\right|{L_F^2(0, \tau ; U)} \leq \mathcal{C}\left|\xi_\tau\right|{L{\mathcal{F}_\tau}^2(\Omega ; O)}$$

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|Formulation of the Problems

$\$ \$$Veft {$$
y_t(t)=A y(t)+B u(t) \quad \text { in }(0, T], y(0)=y_0,
$$\正确的。 \ \$$

## 统计代写|随机过程代写stochastic process代考|Well-Posedness of Stochastic Systems

$\$ \$$\sqrt{1} 左 { d \xi(t)=-\left(A^* \xi(t)+\mathcal{F}(t)^* \xi(t)+\mathcal{G}(t)^* \Xi(t)\right) d t+\Xi(t) d W(t) in [0, \tau), \xi(\tau)=\xi_\tau 正确的。 \operatorname{In}(7.14), \ \tau \in[0, T] \ a n d \ \xi_\tau \in L_{\mathcal{F}\tau}^2(\Omega ; O) \$$. Recallthatweassume $\$ \mathbb{F}$\$isthenatur $|(\backslash x i, \backslash X i)|\left{L{-} F^{\wedge} 2(\backslash O m e g a ; C([0, \backslash t a u] ; O)) \backslash\right.$ times L_F^2\left(0, Itau ; Imathcal ${\mathrm{L}} 2(V$;
O)\right)} \leq \mathcal${C}(\backslash m a t h c a l{F}, \backslash m a t h c a \mid{G}) \backslash \backslash e f t|\backslash x i \backslash t a u \backslash r i g h t|$
$\left{\mathrm{L}{\text { mathcal{F}\gamma }}^{\wedge} 2(\backslash O m e g a ; O)\right}$
$\backslash \lim {\mathrm{n} \backslash$ rightarrow $\backslash$ infty $}$ u_n=u \quad $\backslash$ text ${$ in $} \backslash m a t h c a \mid{U} T \backslash t e x t{$.

## 有限元方法代写

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## 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。