### 金融代写|利率建模代写Interest Rate Modeling代考|MATH4513

Vasicek利率模型一词是指一种对利率的运动和演变进行建模的数学方法。它是一种基于市场风险的单因素短利率模型。瓦西克利率模型常用于经济学中，以确定利率在未来的移动方向。

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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|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. $E\left[X_{n} \mid X_{m}\right]=X_{m}, m \leq n$.
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 continuoustime 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.

Theorem 1.1.1 (The Lundeberg-Levi Central Limit Theorem). For the continuous process, $\bar{X}(t)$, there is
$$\lim {\Delta t \rightarrow 0} P{\bar{X}(s+t)-\bar{X}(s) \leq x}=\frac{1}{\sqrt{2 \pi t}} \int{-\infty}^{x} \exp \left(-\frac{u^{2}}{2 t}\right) d u .$$
Proof: The proof is a matter of applying the central limit theorem. Without loss of generality, we let $s=0$ and take $\Delta t=t / n$. Apparently, there are $\bar{X}(0)=X_{0}, \bar{X}(t)=X_{n}$, and
\begin{aligned} P\left{X_{n}-X_{0} \leq x\right} &=P\left{\sum_{i=1}^{n}\left(X_{i}-X_{i-1}\right) \leq x\right} \ &=P\left{\frac{1 / n \sum_{i=1}^{n}\left(X_{i}-X_{i-1}\right)-0}{\sqrt{\Delta t / n}} \leq \frac{x}{\sqrt{n \Delta t}}\right} \ &=P\left{\frac{1 / n \sum_{i=1}^{n}\left(X_{i}-X_{i-1}\right)-0}{\sqrt{\Delta t / n}} \leq \frac{x}{\sqrt{t}}\right} \end{aligned}

## 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 ws, which are socalled 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 1.1.1. A Brownian motion or a Wiener process is a real-value stochastic process, $W_{t}$ or $W(t), 0 \leq t \leq \infty$, that has the following properties:

1. $W(0)=0$.
1. $W(t+s)-W(t)$ is independent of ${W(u), 0 \leq u \leq t}$.
2. For $t \geq 0$ and $s>0$, the increment $W(t+s)-W(t) \sim N(0, s)$.
3. $W(t)$ 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 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.

We now define the class of functions of stochastic processes such that their values at time $t$ can be determined based on available information up to time $t$. Formally, we introduce the notion of filtration.

Definition 1.1.2. Let $\mathcal{F}{t}$ denote the smallest $\sigma$-algebra containing all sets of the form $$\left{\omega ; W{t_{1}}(\omega) \in B_{1}, \ldots, W_{t_{k}}(\omega) \in B_{k}\right} \subset \Omega,$$
where $k=1,2, \ldots, t_{j} \leq t$ and $B_{j} \subset \boldsymbol{R}$ are Borel sets, where $\boldsymbol{R}$ stands for the set of real numbers. Denote the $\sigma$-algebra as $\mathcal{F}{t}=\sigma(W(s), 0 \leq s \leq t)$; we call the collection of $\left(\mathcal{F}{t}\right)_{t \geq 0}$ a Brownian filtration.

For applications in mathematical finance, it suffices to think of $\mathcal{F}{t}$ as “information up to time $t$ ” or “history of $W{s}$ up to time $t$.” According to the definition, $\mathcal{F}{s} \subset \mathcal{F}{t}$ for $s \leq t$, meaning that a filtration is an increasing stream of information. Readers can find thorough discussions of Brownian filtration in many previous works, for example, Øksendal $(1992)$.
Definition 1.1.3. A function, $f(t)$, is said to be $\mathcal{F}_{t}$-adaptive if
$$f(t)=\vec{f}({W(s), 0 \leq s \leq t}, t)+\quad \forall t$$
that is, the value of the function at time t depends only on the path history up to time $t$.

Adaptive functions ${ }^{1}$ are natural candidates to work with in finance. Suppose that the value of a function represents a decision in investment. Then, such a decision has to be made based on the available information up to the moment of making the decision. The next example gives a good idea of what kind of function is or is not an $\mathcal{F}{t^{-a d a p t i v e}}$ function. Example 1.1 Function $$f(t)= \begin{cases}0 & \min {0 \leq s \leq t} W(s)<2 \ 1 & \min {0 \leq s \leq t} W(s) \geq 2\end{cases}$$ is $\mathcal{F}{t}$-adaptive, whereas
$$f(t)= \begin{cases}0 & \min {0 \leq s \leq 1} W(s)<2 \ 1 & \min {0 \leq s \leq 1} W(s) \geq 2\end{cases}$$
is not $\mathcal{F}_{t}$-adaptive, because $f(t)$ cannot be determined at any time $t<1$.

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

X0=0, Xn+1={Xn−Δ吨,p=12 Xn+Δ吨,1−p=12

1. 的增量Xn+1−Xn独立于\left{X_{i}\right}, \forall i \leq n\left{X_{i}\right}, \forall i \leq n.
2. 和[Xn∣X米]=X米,米≤n.
简单随机游走的一个有趣特征是X一世的时间变化：给定X0, 的方差X一世等于一世Δ吨, 时间序列从X0至X一世.

X¯(吨)=X一世+吨−一世Δ吨Δ吨(X一世+1−X一世),吨∈[一世Δ吨,(一世+1)Δ吨]

\begin{对齐} P\left{X_{n}-X_{0} \leq x\right} &=P\left{\sum_{i=1}^{n}\left(X_{i}-X_ {i-1}\right) \leq x\right} \ &=P\left{\frac{1 / n \sum_{i=1}^{n}\left(X_{i}-X_{i- 1}\right)-0}{\sqrt{\Delta t / n}} \leq \frac{x}{\sqrt{n \Delta t}}\right} \ &=P\left{\frac{1 / n \sum_{i=1}^{n}\left(X_{i}-X_{i-1}\right)-0}{\sqrt{\Delta t / n}} \leq \frac{x }{\sqrt{t}}\right} \end{对齐}\begin{对齐} P\left{X_{n}-X_{0} \leq x\right} &=P\left{\sum_{i=1}^{n}\left(X_{i}-X_ {i-1}\right) \leq x\right} \ &=P\left{\frac{1 / n \sum_{i=1}^{n}\left(X_{i}-X_{i- 1}\right)-0}{\sqrt{\Delta t / n}} \leq \frac{x}{\sqrt{n \Delta t}}\right} \ &=P\left{\frac{1 / n \sum_{i=1}^{n}\left(X_{i}-X_{i-1}\right)-0}{\sqrt{\Delta t / n}} \leq \frac{x }{\sqrt{t}}\right} \end{对齐}

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

1. 在(0)=0.
1. 在(吨+s)−在(吨)独立于在(在),0≤在≤吨.
2. 为了吨≥0和s>0, 增量在(吨+s)−在(吨)∼ñ(0,s).
3. 在(吨)几乎肯定是连续的（as）。
这里ñ(0,s)代表均值为零且方差为 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，1980 a 或 Ikeda 和 Watanabe，1989）。一个示例路径在(吨)如图 1.1 所示，它的生成步长为Δ吨=2−10.

\左{\欧米茄; W{t_{1}}(\omega) \in B_{1}, \ldots, W_{t_{k}}(\omega) \in B_{k}\right} \subset \Omega,\左{\欧米茄; W{t_{1}}(\omega) \in B_{1}, \ldots, W_{t_{k}}(\omega) \in B_{k}\right} \subset \Omega,

F(吨)=F→(在(s),0≤s≤吨,吨)+∀吨

F(吨)={0分钟0≤s≤吨在(s)<2 1分钟0≤s≤吨在(s)≥2是F吨-自适应，而

F(吨)={0分钟0≤s≤1在(s)<2 1分钟0≤s≤1在(s)≥2

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