### 金融代写|利率建模代写Interest Rate Modeling代考|ACTL90003

statistics-lab™ 为您的留学生涯保驾护航 在代写利率建模Interest Rate Modeling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。我们的专家在代写利率建模Interest Rate Modeling代写方面经验极为丰富，各种代写利率建模Interest Rate Modeling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

## 金融代写|利率建模代写Interest Rate Modeling代考|Simple Random Walks

Simple random walks are discrete time series, $\left{X_{i}\right}$, defined as
\begin{aligned} X_{0} &=0, \ X_{n+1} &= \begin{cases}X_{n}-\sqrt{\Delta t}, & p=\frac{1}{2} \ X_{n}+\sqrt{\Delta t}, & 1-p=\frac{1}{2}\end{cases} \end{aligned}
where $\Delta t>0$ stands for the interval of time for stepping forward. One can verify that $\left{X_{i}\right}$ have the following properties:

1. The increment of $X_{n+1}-X_{n}$ is independent of $\left{X_{i}\right}, \forall i \leq n$.
2. $E\left[X_{n} \mid X_{m}\right]=X_{m}, m \leq n$.
3. $\operatorname{Var}\left[X_{n} \mid X_{m}\right]=(n-m) \Delta t, m \leq n$.
An interesting feature of the simple random walk is the linearity of $X_{i}$ ‘s variance in time: given $X_{0}$, the variance of $X_{i}$ is equal to $i \Delta t$, the time it takes the time series to evolve from $X_{0}$ to $X_{i}$.

Out of the simple Brownian random walk, we can construct a continuous-time process through linear interpolation:
$$\bar{X}(t)=X_{i}+\frac{t-i \Delta t}{\Delta t}\left(X_{i+1}-X_{i}\right), \quad t \in[i \Delta t,(i+1) \Delta t]$$
We are interested in the limiting process of $\bar{X}(t)$ as $\Delta t \rightarrow 0$, in the hope that the limit remains a meaningful stochastic process. The next theorem confirms just that.

## 金融代写|利率建模代写Interest Rate Modeling代考|Brownian Motion

A continuous stochastic process is a collection of real-valued random variables, ${X(t, \omega), 0 \leq t \leq T}$ or $\left{X_{t}(\omega), 0 \leq t \leq T\right}$, that are defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Here $\Omega$ is the collection of all $\omega$ s, which are so-called sample points, $\mathcal{F}$ the smallest $\sigma$-algebra that contains $\Omega$, and $\mathbb{P}$ a probability measure on $\Omega$. Each random outcome, $\omega \in \Omega$, corresponds to an entire time series
$$t \rightarrow X_{t}(\omega), \quad t \in T$$
which is called a path of $X_{t}$. In view of Equation 1.7, we can regard $X_{t}(\omega)$ as a function of two variables, $\omega$ and $t$. For notational simplicity, however, we often suppress the $\omega$ variable when its explicit appearance is not necessary.

In the context of financial modeling, we are particularly interested in the Brownian motion introduced earlier. Its formal definition is given below.

Definition 1.1 A Brownian motion or a Wiener process is a realvalued stochastic process, $W_{t}$ or $W(t), 0 \leq t \leq \infty$, that has the following properties:

1. $W(0)=0$.
2. $W(t+s)-W(t)$ is independent of ${W(u), 0 \leq u \leq t}$.
3. For $t \geq 0$ and $s>0$, the increment $W(t+s)-W(t) \sim N(0, s)$.
4. $W(t)$ is continuous almost surely (a.s.).
Here $N(0, s)$ stands for a normal distribution with mean zero and variance s. Note that in some literature, property 4 is not part of the definition, as it can be proved to be implied by the first three properties (Varadhan, $1980 \mathrm{a}$ or Ikeda and Watanabe, 1989). A sample path of $W(t)$ is shown in Figure $1.1$, which is generated with a step size of $\Delta t=2^{-10}$.

Brownian motion plays a major role in continuous time stochastic modeling in physics, engineering and finance. In finance, it has been used to model the random behavior of asset returns. Several major properties of Brownian motion are listed below.

## 金融代写|利率建模代写Interest Rate Modeling代考|STOCHASTIC INTEGRALS

Stochastic calculus considers the integration and differentiation of general $\mathcal{F}{t}$-adaptive functions. The purpose of developing such a stochastic calculus is to model financial time series (with random dynamics) with either integral or differential equations. According to Lemma 1.1, a Brownian motion, $W(t)$, is nowhere differentiable in the usual sense of differentiation for deterministic functions. To define differentials of stochastic processes in a proper sense, we must first study the notion of stochastic integrals. Stochastic integrals can be defined for functions in the square-integrable space, $H^{2}[0, T]=L^{2}(\Omega \times[0, T], \mathrm{d} \mathbb{P} \times \mathrm{d} t)$, which is defined to be the collection of functions satisfying $$E\left[\int{0}^{T}|f(t, \omega)|^{2} \mathrm{~d} t\right]<\infty$$
Note that, without indicated otherwise, $E[\cdot]$ means $E^{\mathbb{P}}[\cdot]$, the unconditional expectation under $\mathbb{P}$. The definition consists of a three-step procedure. First, we make the definition for elementary or piecewise constant functions in an intuitive way. Second, we define the integrals of a bounded continuous function as a limit of integrals of elementary functions. Finally, we define the integral of a general square-integrable function as a limit of integrals of bounded continuous functions. The key in this three-step procedure is of course to ensure the convergence of the limits in $L^{2}(\Omega, \mathcal{F}, \mathbb{P})$, the Hilbert space of random variables satisfying
$$E\left[X^{2}(\omega)\right]<\infty$$
This definition approach is taken by Oksendal (1992). Alternative treatments of course also exist; see, for example, Mikosch (1998).

## 金融代写|利率建模代写Interest Rate Modeling代考|Simple Random Walks

$$X_{0}=0, X_{n+1} \quad=\left{X_{n}-\sqrt{\Delta t}, \quad p=\frac{1}{2} X_{n}+\sqrt{\Delta t}, \quad 1-p=\frac{1}{2}\right.$$

1. 的增量 $X_{n+1}-X_{n}$ 独立于⿴left $\left{X_{-}{i} \backslash\right.$ Iight}, Iforall i leq n.
2. $E\left[X_{n} \mid X_{m}\right]=X_{m}, m \leq n$.
3. $\operatorname{Var}\left[X_{n} \mid X_{m}\right]=(n-m) \Delta t, m \leq n$.
简单随机游走的一个有趣特征是 $X_{i}$ 的时间变化: 给定 $X_{0}$, 的方差 $X_{i}$ 等于 $i \Delta t$, 时间序列从 $X_{0}$ 至 $X_{i}$.
从简单的布朗随机游走中，我们可以通过线性揷值构造一个连续时间的过程:
$$\bar{X}(t)=X_{i}+\frac{t-i \Delta t}{\Delta t}\left(X_{i+1}-X_{i}\right), \quad t \in[i \Delta t,(i+1) \Delta t]$$
我们对限制过程感兴趣 $\bar{X}(t)$ 作为 $\Delta t \rightarrow 0$ ，希望极限仍然是一个有意义的随机过程。下一个定理证实了这一点。

## 金融代写|利率建模代写Interest Rate Modeling代考|Brownian Motion

$$t \rightarrow X_{t}(\omega), \quad t \in T$$

1. $W(0)=0$.
2. $W(t+s)-W(t)$ 独立于 $W(u), 0 \leq u \leq t$.
3. 为了 $t \geq 0$ 和 $s>0$, 增量 $W(t+s)-W(t) \sim N(0, s)$.
4. $W(t)$ 几平肯定是连续的 (as)。
这里 $N(0, s)$ 代表均值为零且方差为 $\mathrm{s}$ 的正态分布。请注意，在某些文献中，属性 4 不是定义的一部分，因 为可以证明前三个属性暗示了它 (Varadhan，1980a或池田和渡边，1989) 。一个示例路径 $W(t)$ 如图1.1 ，它的生成步长为 $\Delta t=2^{-10}$.
布朗运动在物理学、工程和金融领域的连续时间随机建模中发挥着重要作用。在金融领域，它已被用于模拟资产回 报的随机行为。下面列出了布朗运动的几个主要性质。

## 金融代写|利率建模代写Interest Rate Modeling代考|STOCHASTIC INTEGRALS

$$E\left[\int 0^{T}|f(t, \omega)|^{2} \mathrm{~d} t\right]<\infty$$

$$E\left[X^{2}(\omega)\right]<\infty$$
Oksendal (1992) 采用了这种定义方法。当然也存在替代疗法；例如，参见 Mikosch (1998)。

## 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。