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## 金融代写|风险估值理论代写Uncertainty Quantification代考|Recession Approach for Linear Variational Inequalities

Let $H$ be a Hilbert space, $a: H \times H \rightarrow \mathbb{R}$ be a positive and continuous bilinear form, $K \subset H$ be a nonempty, closed, and convex set, and $f \in H^*$.
We continue to consider the variational inequality: Find $u \in K$ such that
$$a(u, v-u) \geq\langle f, v-u\rangle, \quad \text { for every } v \in K .$$
Let $\mathcal{S}(a, f, K)$ be the set of solutions of (4.26).
We formulate the following:
Assumption (A): There are two linear maps $P_0: H \rightarrow H$ and $P_1: H \rightarrow H$ such that the set $P_0(K)$ is bounded, $P_1$ is compact, and there is a constant $\alpha>0$ such that
$$a(v, v)+\left|P_0 v\right|^2+\left|P_1 v\right|^2 \geq \alpha|v|^2, \quad \text { for every } v \in H .$$
Note that due to (4.27), for continuous and positive bilinear form $a(\cdot, \cdot)$, the term $\left[a(v, v)+\left|P_0 v\right|^2+\left|P_1 v\right|^2\right]^{1 / 2}$ defines a norm which is equivalent to the original norm. Evidently, Assumption (A) is satisfied trivially if either $a(\cdot, \cdot)$ is elliptic or $K$ is bounded.
The following notion of a recession cone will play a central role:
Definition 4.2.1 Let $H$ be a Hilbert space and $C \subset H$ be a closed, and convex set. The recession cone of $C$, denoted by $C_{\infty}$, is defined, for any fixed $c_0 \in C$, by
$$C_{\infty}:=\bigcap_{t>0} t\left(C-c_0\right) .$$

## 金融代写|风险估值理论代写Uncertainty Quantification代考|Existence Results and Stability of Solutions

Let $H$ be a Hilbert space with inner product $\langle\cdot, \cdot\rangle$ and $|\cdot|, K \subset H$ be closed and convex, $F: H \rightarrow H$ be a given map, and $f \in H$.
We consider the nonlinear variational inequality of finding $u \in K$ such that
$$\langle F u-f, v-u\rangle \geq 0, \quad \text { for every } v \in K .$$
Let $\mathcal{S}(F, f, K)$ be the solution set of (4.34).
By using the indicator function, (4.34) can be written as a generalized equation:
$$f \in F u+\partial I_K u,$$
or equivalently, by using the normal cone of $K$ at $u$ :
$$f-F u \in N_K u .$$
In Theorem 4.2.1, we characterized linear variational inequality (4.10) as an equivalent optimization problem. A similar equivalence exists for nonlinear variational inequalities, albeit under more restrictive conditions, as shown next:

Theorem 4.3.1 Let $H$ be a Hilbert space, $K \subset H$ be closed, and convex, and $f \in$ H. Let $F: K \rightarrow H$ be monotone, continuous, and potential. Then variational inequality (4.34) is equivalent to the minimization problem: Find $u \in K$ such that
$$\Phi(u) \leq \Phi(v), \quad \text { for every } v \in K$$

where, for some fixed $\tilde{v} \in K, \Phi: H \rightarrow \mathbb{R}$ is a convex function defined by
$$\Phi(v)=\int_0^1\langle F(\tilde{v}+\tau(v-\tilde{v})), v-\tilde{v}\rangle d \tau-\langle f, v\rangle+\Phi_0$$
with $\Phi_0=\Phi(\tilde{v})+\langle f, \tilde{v}\rangle$, and it satisfies the following identity:
$$\nabla \Phi(u)=F u-f, \quad \text { for every } u \in K$$
Proof. Since $F$ is potential, there is a functional $\Psi$ such that $F=\nabla \Psi$. Therefore, $\left.\frac{d}{d t} \Psi(\tilde{v}+t(v-\tilde{v}))\right|_{t=\tau}=\langle\nabla \Psi(\tilde{v}+\tau(v-\tilde{v})), v-\tilde{v}\rangle=\langle F(\tilde{v}+\tau(v-\tilde{v})), v-\tilde{v}\rangle$, and by integrating in $\tau$ from 0 to 1 , we obtain
$$\Psi(v)-\Psi(\tilde{v})=\int_0^1\langle F(\tilde{v}+\tau(v-\tilde{v})), v-\tilde{v}\rangle d \tau .$$
Defining $\Phi(\cdot)=\Psi(\cdot)-\langle f, \cdot\rangle$, we obtain (4.36) or equivalently (4.37). Since $F$ is monotone, $\nabla \Phi$ is monotone, which is equivalent to the convexity of $\Phi$. The equivalence of (4.34) and (4.35) is now quite evident.

## 金融代写|风险估值理论代写Uncertainty Quantification代考|Recession Approach for Linear Variational Inequalities

$$a(u, v-u) \geq\langle f, v-u\rangle, \quad \text { for every } v \in K \text {. }$$

$$C_{\infty}:=\bigcap_{t>0} t\left(C-c_0\right) .$$

## 金融代写川风险估值理论代写Uncertainty Quantification代考|Existence Results and Stability of Solutions

$$\langle F u-f, v-u\rangle \geq 0, \quad \text { for every } v \in K .$$

$$f \in F u+\partial I_K u$$

$$f-F u \in N_K u .$$

$$\Phi(u) \leq \Phi(v), \quad \text { for every } v \in K$$

$$\Phi(v)=\int_0^1\langle F(\bar{v}+\tau(v-\bar{v})), v-\bar{v}\rangle d \tau-\langle f, v\rangle+\Phi_0$$

$$\nabla \Phi(u)=F u-f, \quad \text { for every } u \in K$$

$$\Psi(v)-\Psi(\bar{v})=\int_0^1\langle F(\bar{v}+\tau(v-\bar{v})), v-\bar{v}\rangle d \tau$$

## 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。

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## 金融代写|风险估值理论代写Uncertainty Quantification代考|Modes of Convergence of Random Variables

In the section, we discuss notions of convergence for random variables.
Definition 2.5.1 Let $H$ be a real Hilbert space and $\left{X_n\right}$ be a sequence of $H$-valued random variables. The sequence $\left{X_n\right}$ is said to converge to $X \in H$

1. almost surely, if $X_n(\omega) \rightarrow X(\omega)$ for almost all $\omega \in \Omega$. That is, if $\mathbb{P}\left(\lim _{n \rightarrow \infty}\left|X_n-X\right|=0\right)=1$
2. in probability if $\mathbb{P}\left(\left|X_n-X\right|>\epsilon\right) \rightarrow 0$ as $n \rightarrow \infty$ for any $\epsilon>0$.
3. in $p$-th mean or in $L^p(\Omega, H)$, for $1 \leq p<\infty$, if $\mathbb{E}\left[\left|X_n-X\right|^p\right] \rightarrow 0$ as $n \rightarrow \infty$. For $p=2$, this convergence is referred to as the mean-square convergence.

Remark 2.5.1 For $q>p$, the convergence in $q$-th mean implies the convergence in $p$-th mean which further implies convergence in probability. Almost surely convergence implies convergence in probability. Furthermore, convergence almost surely implies convergence in $p$-th mean if the random variables are uniformly bounded.
Definition 2.5.2 A set of random variables $\left{X_1, X_2, \ldots\right}$ is called independent and identically distributed (iid, for short) if each $X_j$ has identical distribution and the distinct random variables $X_i, X_j$, for $i \neq j$, are independent.

Theorem 2.5.1 (Strong Law of Large Numbers.) Let $\left{X_1, X_2, \ldots, X_M\right}$ be $M$ iid real-valued random variable with mean $\mu$ such that $\mathbb{E}\left[\left|X_j\right|\right]<\infty$. Then, the sample mean $\bar{X}_M$ of $X_1, X_2, \ldots, X_M$, defined by
$$\bar{X}_M=\frac{X_1+X_2+\cdots+X_M}{M},$$
converges to $\mu$, almost surely, as $M \rightarrow \infty$.

## 金融代写|风险估值理论代写Uncertainty Quantification代考|Projections on Convex Sets in Hilbert Spaces

Let $H$ be a Hilbert space with the inner product $\langle\cdot, \cdot\rangle$ and norm $|\cdot|=\sqrt{\langle\cdot, \cdot}$. Let $H^*$ be the dual of $H$ which, by the Riesz isomorphism, will be identified with $H$. For simplicity, we only consider real Hilbert/Banach spaces. Our focus is on the projection map which is formally defined as follows:

Definition 3.1.1 Let $H$ be a Hilbert space and $K$ be a nonempty, closed, and convex subset of $H$. The projection map $P_K: H \rightarrow K$ assigns to any $x \in H$, the unique point $P_K x$ in $K$ which is closest to $x$. The point $P_K x$ is called the projection of $x$ onto K. That is,
$$\left|x-P_K x\right| \leq|x-z|, \text { for every } z \in K .$$
In the following result, besides showing that the projection map is well-defined, we give its important equivalent variational characterization:

Theorem 3.1.1 Let $H$ be a Hilbert space, and $K$ be a nonempty, closed, and convex subset of H. Let $x \in H$. Then there exists a unique $P_K x \in K$ such that
$$\left|x-P_K x\right| \leq|x-z|, \text { for every } z \in K .$$
Moreover, $P_K x$ is the projection of $x$, if and only if, it satisfies
$$\left\langle x-P_K x, z-P_K x\right\rangle \leq 0, \text { for every } z \in K .$$
Proof. Since
$$\delta:=\inf {z \in K}|x-z| \geq 0,$$ there is a minimizing sequence $\left{y_n\right}$ in $K$ with $\left|x-y_n\right| \rightarrow \delta$, as $n \rightarrow \infty$. Evidently, the sequence $\left{y_n\right}$ is bounded in $H$, and by the reflexivity of $H$, there exists a weakly convergent subsequence. By keeping the same notation for subsequences as well, let $\left{y_n\right}$ be the subsequence that converges weakly to some $y \in H$. The set $K$ being closed and convex is also weakly closed, and hence $y \in K$. Since any norm is a weakly lower semi-continuous function, we obtain $$|x-y| \leq \liminf {n \rightarrow \infty}\left|x-y_n\right|=\delta,$$
which in view of the inequality $\delta \leq|x-y|$ confirms that indeed $|x-y|=\delta$. This proves the existence of an element $y:=P_K x \in K$ that satisfies (3.1).

## 五融代写|风险估值理论代写Uncertainty Quantification代考|Modes of Convergence of Random Variables

\left 的分隔符缺失或无法识别 是一个序列 $H$ 值随机㚆量。序列

$$\bar{X}_M=\frac{X_1+X_2+\cdots+X_M}{M},$$

## 金融代写|风险估值理论代与写certainty Quantification代考|Projections on

Convex Sets in Hilbert Spaces 考慮真正的 Hilbert/Banach 空间。我们的重点是投影图，其正式定义如下:

$$\left|x-P_K x\right| \leq|x-z| \text {, for every } z \in K \text {. }$$

$$\left|x-P_K x\right| \leq|x-z| \text {, for every } z \in K \text {. }$$

$$\left\langle x-P_K x, z-P_K x\right\rangle \leq 0 \text {, for every } z \in K .$$

$$\delta:=\inf z \in K|x-z| \geq 0,$$

\left 的分隔符缺失或无法识别 有界 $H$ ，并且通过自反性 $H$ ，存在弱收玫子序列。通过对子序列也保持相同的 符号，让 left 的分隔符缺失或无法识别 是弱收敛到某个的子序列 $y \in H$. 套装 $K$ 封闭和凸也也是弱封闭的，因 此 $y \in K$. 由于任何范数都是弱下半连续函数，我们得到
$$|x-y| \leq \liminf n \rightarrow \infty\left|x-y_n\right|=\delta,$$

## 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 Categories:Uncertainty Quantification, 金融代写, 风险估值理论

## avatest™帮您通过考试

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## 金融代写|风险估值理论代写Uncertainty Quantification代考|Elements of Functional Analysis

Definition 1.1.1 A vector space $X$ over a scalar field $\mathbb{F}$ is an assemblage of elements, called vectors, that is closed under two algebraic operations, namely, vector addition and multiplication by scalars, that is, for any $u, v \in X$ and $\alpha \in \mathbb{F}$, we have $u+v \in X$ and $\alpha u \in X$. The algebraic operations satisfy the following:

1. For all $u, v \in X$, we have $u+v=v+u$.
2. For all $u, v, w \in X$, we have $u+(v+w)=(u+v)+w$.
3. There exists a vector, denoted by 0 , such that $u+0=u$, for any $u \in X$.
4. For any $u \in X$, there is a unique $-u \in X$ such that $u+(-u)=0$.
5. For any $\alpha, \beta \in \mathbb{F}$ and $u \in X$, we have $\alpha(\beta u)=(\alpha \beta) u$.
6. For the scalar 1 of $F$, we have $1 u=u$, for any $u \in X$.
7. For all $\alpha \in \mathbb{F}$ and $u, v \in X$, we have $\alpha(u+v)=\alpha u+\alpha v$.
8. For all $\alpha, \beta \in \mathbb{F}$ and $u \in X$, we have $(\alpha+\beta) u=\alpha u+\beta u$.

The vector space $X$ is called a real vector space, if $\mathbb{F}=\mathbb{R}$. The vector space $X$ is called a complex vector space, if $\mathbb{F}=\mathrm{C}$.

## 金融代写|风险估值理论代写Uncertainty Quantification代考|Fundamentals of Measure Theory and Integration

We now present an overview of the key ideas of measure theory and integration.
Definition 1.2.1 Let $X$ be a given set and $\mathcal{F}$ be a collection of subsets of $X$. The collection $\mathcal{F}$ is called a $\sigma$-algebra, if the following conditions hold:

1. $\emptyset \in \mathcal{F}$.
2. If $F \in \mathcal{F}$, then its complement $F^c:={x \in X \mid x \notin F} \in \mathcal{F}$.
3. If $F_j \in \mathcal{F}$, for $j=1,2, \ldots$, then $\bigcup_{j=1}^{\infty} F_j \in \mathcal{F}$.
If $X$ is a topological space, then the Borel $\sigma$-algebra, denoted by $\mathcal{B}(X)$, is the smallest $\sigma$-algebra containing all open subsets of $X$.

It is evident from the above definition that if $\mathcal{F}$ is a $\sigma$-algebra, then $X \in \mathcal{F}$, and if, $F, G \in \mathcal{F}$, then $F \backslash G=F \cap G^c \in \mathcal{F}$. Moreover, if $F_i \in \mathcal{F}$, for $i=1,2, \ldots$, then $\bigcap_{j=1}^{\infty} F_j \in \mathcal{F}$. The pair $(X, \mathcal{F})$ is termed as a measurable space and any set $F \in \mathcal{F}$ is called a measurable set.

Definition 1.2.2 A measure on a measurable space $(X, \mathcal{F})$ is a function $\mu: \mathcal{F} \rightarrow$ $[0,+\infty]$ satisfying the following conditions:

1. $\mu(\emptyset)=0$.
2. $\mu\left(\bigcup_{i=1}^{\infty} F_i\right)=\sum_{i=1}^{\infty} \mu\left(F_i\right)$, if $F_i \cap F_j=\emptyset$, for $i \neq j$.
The triple $(X, \mathcal{F}, \mu)$ is called a measure space. The measure space is called $\sigma$-finite, if $X=\cup_{j=1}^{\infty} F_j$, for $F_j \in \mathcal{F}$ with $\mu\left(F_j\right)<\infty$. A measure space $(X, \mathcal{F}, \mu)$ is called a complete measure space, if $F \in \mathcal{F}$ with $\mu(F)=0$ and $G \subset F$, then $G \in \mathcal{F}$. The set $F \in \mathcal{F}$ with $\mu(F)=0$ is called a null set.

## 金融代写|风险估值理论代写Uncertainty Quantification代考|Fundamentals of Measure Theory and Integration

㧴们现在概述则度论和集成的关键思想。

$\emptyset \in \mathcal{F}$.

$\mu(\emptyset)=0$.

$\mu\left(\bigcup_{i=1}^{\infty} F_i\right)=\sum_{i=1}^{\infty} \mu\left(F_i\right)$, 如果 $F_i \cap F_j=\emptyset$, 为了 $i \neq j$.

## 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。