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

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• 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|Random Variables

A random variable $X$ is defined as a real-valued measurable function on a sample space $\Omega$. We regard interest rates, bond prices, and stock prices on future days as random variables throughout this book.
The expectation of a random variable $X$ under measure $\mathbf{P}$ is defined as
$$E[X]=\int_{w \in \Omega} X(w) \mathbf{P}(d w)$$
The expectation is sometimes referred to as the mean. The variance of $X$ is defined as
$$v(x)=\int_{w \in \Omega} X(w)^{2} \mathbf{P}(d w)$$
The variance $V(X)$ is a measure of how widely $X$ varies from $E[X]$.
Expectation is linear on linear combinations of random variables. Let $Y$ be another random variable. As an example, for constants $a, b$, and $c$, the expectation of the linear combination $a+b X+c Y$ is given by
$$E[a+b X+c Y]=a+b E[X]+c[Y]$$
The variance of $a+b X+c Y$ is given by
$$V(a+b X+c Y)=b^{2} V(X)+c^{2} V(Y)+2 b c \operatorname{Cov}(X, Y)$$

where Cov is the covariance between $X$ and $Y$ and is defined as
$$\operatorname{Cov}(X, Y)=\int_{w \in \Omega}(X(w)-E[X])(Y(w)-E[Y]) P(d w)$$
The correlation coefficient $\rho(X, Y)$ is defined as
$$\rho(X, Y)=\frac{\operatorname{Cov}(X, Y)}{\sqrt{V(X) V(Y)}}$$
For any constants $a$ and $b$, it holds that
\begin{aligned} \rho(a X, b Y) &=\frac{\operatorname{Cov}(a X, b Y)}{\sqrt{V(a X) V(b Y)}}=\frac{a b \operatorname{Cov}(X, Y)}{\sqrt{a^{2} b^{2} V(X) V(Y)}} \ &=\rho(X, Y) . \end{aligned}
Hence, the correlation coefficient is invariant to the magnitudes of $a$ and $b$. The correlation coefficient can be used as a measure of the strength of the linear relationship between $X$ and $Y$. Naturally, it holds that $-1 \leq \rho \leq 1$.

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

For arbitrage pricing, the dynamics of market price change is represented by a stochastic process. To introduce the concept of a stochastic process, we develop a probability space such that stochastic processes are well defined in the space, where the key is a filtration used to define a stochastic process.
Filtration
For a fixed time $\tau>0$, let $\mathcal{F}{t \in[0, \tau]}$ denote an increasing subset of $\mathcal{F}$; that is, $\mathcal{F}{s} \subset \mathcal{F}{t}$ for $0 \leq s{t}$ is a $\sigma$-algebra. Under this definition, $\mathcal{F}{t \in[0, \tau]}$ is called an augmented filtration, or a filtration for short. Along these lines, $\left(\Omega, \mathcal{F}, \mathcal{F}{t \in[0, \tau]}, \mathbf{P}\right)$ is called a filtered probability space, or a probability space for short.
Without loss of generality, we may set $\mathcal{F}=\mathcal{F}{\tau}$, which means that $\left(\Omega, \mathcal{F}, \mathcal{F}{t \in[0, \tau]}, \mathbf{P}\right)$ can be represented by a triplet $\left(\Omega, \mathcal{F}{t \in[0, \tau]}, \mathbf{P}\right)$ for convenience. In particular, we use a filtration chosen such that it represents a flow of information; namely, $\mathcal{F}{t}$ comprises the information about all events observed up to time $t$. An example of such a filtration is given in Example 2.3.1.
Stochastic process
Recall the probability space introduced in Example 2.1.1, that is, the up and down moves of stock price. Stock price movement between days is uncertain and unpredictable, making it an example of a discrete-time stochastic process.
In a continuous-time setting, a stochastic process $X_{t}$ is defined as a set of random variables indexed by time $t, t \geq 0$ and defined on a probability space $\left(\Omega, \mathcal{F}{t \in[0, \tau]}, \mathbf{P}\right)$. The process $X{t}$ is said to be continuous if almost all sample paths are continuous in $t$. Additionally, $X_{t}$ is said to be adapted if $X_{t}$ is $\mathcal{F}{t^{-}}$ measurable at an arbitrary time $t \in[0, \tau]$. This is called $\mathcal{F}{t}$-adapted for short. In this book, we work mostly with adapted processes.

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

This section summarizes basic definitions and results for stochastic differential equations but omits proofs. For details and a rigorous description see any of Chung and Williams (1990), Karatzas and Shreve (1998), Öksendal (2003), and Shreve (2004).
Stochastic integral
Let $W_{t}$ be a Brownian motion under a measure $\mathbf{P}$. For a stochastic process $X_{t}$ satisfying $\int_{0}^{t}\left|X_{s}\right|^{2} d s<\infty$ a.s., the stochastic integral of $X_{t}$ with respect to $W_{t}$ is defined by
$$\int_{0}^{t} X_{s} d W_{s}=\lim {\Delta \rightarrow 0} \sum{i=1}^{n-1} X_{t_{i}}\left{W_{t_{i+1}}-W_{t_{i}}\right}$$
where $0=t_{0}<\cdots<t_{n}=t$ and $\Delta t=\max {0 \leq i \leq n-1}\left(t{i+1}-t_{i}\right)$.
The following properties are well known.
1) If $X_{t}$ is a deterministic process, that is, generated by a deterministic function on $t$, then it holds that
$$E\left[\int_{0}^{t} X_{s} d W_{s}\right]=0$$
2) $\int_{0}^{t} X_{s} d W_{s}$ has normal distribution with mean 0 and variance
$$E\left[\left(\int_{0}^{t} X_{s} d W_{s}\right)^{2}\right]=\int_{0}^{t}\left|X_{s}\right|^{2} d s$$
Generally, for a function $\sigma\left(t, X_{t}\right)$ with inputs $t$ and $X_{t}$, and regarding $\sigma\left(t, X_{t}\right)$ as a stochastic process, the stochastic integral is defined by
$$\int_{0}^{t} \sigma\left(s, X_{s}\right) d W_{s}$$

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

ρ(X,是)=这⁡(X,是)在(X)在(是)

ρ(一个X,b是)=这⁡(一个X,b是)在(一个X)在(b是)=一个b这⁡(X,是)一个2b2在(X)在(是) =ρ(X,是).

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

Let在吨是一个测度下的布朗运动磷. 对于随机过程X吨令人满意的∫0吨|Xs|2ds<∞作为，随机积分X吨关于在吨定义为

\int_{0}^{t} X_{s} d W_{s}=\lim {\Delta \rightarrow 0} \sum{i=1}^{n-1} X_{t_{i}}\left {W_{t_{i+1}}-W_{t_{i}}\right}\int_{0}^{t} X_{s} d W_{s}=\lim {\Delta \rightarrow 0} \sum{i=1}^{n-1} X_{t_{i}}\left {W_{t_{i+1}}-W_{t_{i}}\right}

1) 如果X吨是一个确定性过程，即由一个确定性函数在吨，那么它认为

2) ∫0吨Xsd在s具有均值为 0 且方差为正态分布

∫0吨σ(s,Xs)d在s

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