19th Ave New York, NY 95822, USA

# 数学代写|随机偏微分方程代写Stochastic Differential Equation代考|MAST31712 Wiener processes

my-assignmentexpert™提供最专业的一站式服务：Essay代写，Dissertation代写，Assignment代写，Paper代写，Proposal代写，Proposal代写，Literature Review代写，Online Course，Exam代考等等。my-assignmentexpert™专注为留学生提供Essay代写服务，拥有各个专业的博硕教师团队帮您代写，免费修改及辅导，保证成果完成的效率和质量。同时有多家检测平台帐号，包括Turnitin高级账户，检测论文不会留痕，写好后检测修改，放心可靠，经得起任何考验！

## 数学代写|随机偏微分方程代写Stochastic Differential Equation代考|Wiener processes

Definition 4.10 Let $q>0$. A real-valued mean-zero Gaussian process $W=$ $(W(t), t \geq 0)$ with continuous trajectories and covariance function
$$\mathbb{E} W(t) W(s)=(t \wedge s) q, \quad t, s \geq 0,$$
is called a Wiener process with diffusion $q$. If the diffusion is equal to 1 then $W$ is called standard.

Definition 4.11 Assume that $\left(\Omega, \mathcal{F},\left(\mathcal{F}_t\right), \mathbb{P}\right)$ is a filtered probability space and that $W$ is a Wiener process in $\mathbb{R}^d$ adapted to $\left(\mathcal{F}_t\right)$. Then $W$ is a Wiener process with respect to $\left(\mathcal{F}_t\right)$ or an $\left(\mathcal{F}_t\right)$-Wiener process if, for all $t, h \geq 0, W(t+h)-W(t)$ is independent of $\mathcal{F}_t$.

We have the following classical Lévy characterization of a Wiener process; see e.g. Kallenberg (2002).

## 数学代写|随机偏微分方程代写Stochastic Differential Equation代考|Compound Poisson processes in a Hilbert space

Definition 4.14 Let $\nu$ be a finite measure on a Hilbert space $U$ such that $\nu({0})=$ 0 . A compound Poisson process with the Lévy measure (also called the jump intensity measure) $v$ is a càdlàg Lévy process $L$ satisfying
$$\mathbb{P}(L(t) \in \Gamma)=\mathrm{e}^{-v(U) t} \sum_{k=0}^{\infty} \frac{t^k}{k !} v^{* k}(\Gamma), \quad \forall t \geq 0, \Gamma \in \mathcal{B}(U) .$$

In the formula above, we use the convention that $v^0$ is equal to the unit measure concentrated at ${0}$, that is, $v^0=\delta_0$.

The theorem below provides the construction of a compound Poisson process with given $\nu$.

Theorem 4.15 Let $v$ be a finite measure supported on $U \backslash{0}$, and let $a=v(U)$.
(i) Let $Z_1, Z_2, \ldots$ be independent random variables with identical laws equal to $a^{-1} v$. In addition, let $(\Pi(t), t \geq 0)$ be a Poisson process with intensity a, independent of $Z_1, Z_2, \ldots$ Then
$$L(t)=\sum_{j=1}^{\Pi(t)} Z_j$$
is a compound Poisson process with jump intensity measure $\nu$.
(ii) Given a compound Poisson process $L$ with jump intensity measure $v$, one can find a sequence of independent random variables $Z_1, Z_2, \ldots$ with identical laws equal to $a^{-1} v$ and a Poisson process $(\Pi(t), t \geq 0)$ with intensity $a$, independent of $Z_1, Z_2, \ldots$, such that (4.6) holds.
(iii) For $z \in \mathbb{C}, t \geq 0$ and $x \in U$,
$$\mathbb{E} \mathrm{e}^{z\langle x, L(t)\rangle_U}=\exp \left{-t \int_U\left(1-\mathrm{e}^{z\langle x, y\rangle_U}\right) v(\mathrm{~d} y)\right}$$

# 随机偏微分方程代写

## 数学代写|随机偏微分方程代写STOCHASTIC DIFFERENTIAL EQUATION代考|WIENER PROCESSES

$$\mathbb{E} W(t) W(s)=(t \wedge s) q, \quad t, s \geq 0,$$

## 数学代写|随机偏微分方程代写STOHASTIC DIFFERENTIAL EQUATION代考|COMPOUND POISSON PROCESSES IN A HILBERT SPACE

$i$ 让 $Z_1, Z_2, \ldots$ 是具有相同规律的独立随机变量等于 $a^{-1} v$. 另外，让 $(\Pi(t), t \geq 0)$ 是强度为 a 的泊松过程，与 $Z_1, Z_2, \ldots$ 然后
$$L(t)=\sum_{j=1}^{\Pi(t)} Z_j$$

ii i为了 $z \in \mathbb{C}, t \geq 0$ 和 $x \in U$,

## 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。