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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

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|THE FORCE OF INTEREST

Equation $2.3 .2$ indicates how $i_h(t)$ is defined in terms of the accumulation factor $A(t, t+h)$. In Example 2.2.3 we gave (in relation to a particular time $t_0$ ) the values of $i_h\left(t_0\right)$ for a series of values of $h$, varying from $1 / 4$ (i.e., 3 months) to $1 / 365$ (i.e., 1 day). The trend of these values should be noted. In practical situations, it is not unreasonable to assume that, as $h$ becomes smaller and smaller, $i_h(t)$ tends to a limiting value. In general, of course, this limiting value will depend on $t$. We therefore assume that for each value of $t$ there is a number $\delta(t)$ such that
$$\lim _{h \rightarrow 0^{+}} i_h(t)=\delta(t)$$
The notation $h \rightarrow 0^{+}$indicates that the limit is considered as $h$ tends to zero “from above”, i.e., through positive values. This is, of course, always true in the limit of a time interval tending to zero.

It is usual to call $\delta(t)$ the force of interest per unit time at time $t$. In view of Eq. 2.4.1, $\delta(t)$ is sometimes called the nominal rate of interest per unit time at time $t$ convertible momently. Although it is a mathematical idealization of reality, the force of interest plays a crucial role in compound interest theory. Note that by combining Eqs $2.3 .2$ and 2.4.1, we may define $\delta(t)$ directly in terms of the accumulation factor as
$$\delta(t)=\lim _{h \rightarrow 0^{+}}\left[\frac{A(t, t+h)-1}{h}\right]$$ The force of interest function $\delta(t)$ is defined in terms of the accumulation function $A\left(t_1, t_2\right)$, but when the principle of consistency holds, it is possible, under very general conditions, to express the accumulation factor in terms of the force of interest. This result is contained in Theorem 2.4.1.

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|PRESENT VALUES

In Section 2.3, accumulation factors were introduced to quantify the growth of an initial investment as time moves forward. However, one can consider the situation in the opposite direction. For example, if one has a future liability of known amount at a known future time, how much should one invest now (at known interest rate) to cover this liability when it falls due? This leads us to the concept of present values.

Let $t_1 \leq t_2$. It follows by Eq. $2.3 .3$ that an investment of $\frac{C}{A\left(t_1, t_2\right)}$, i.e., $C \exp \left(-\int_{t_1}^{t_2} \delta(t) d t\right)$, at time $t_1$ will produce a return of $C$ at time $t_2$. We therefore say that the discounted value at time $t_1$ of $C$ due at time $t_2$ is
$$C \exp \left[-\int_{L_1}^{t_2} \hat{\delta}(t) \mathrm{d} t\right]$$
This is the sum of money which, if invested at time $t_1$, will give $C$ at time $t_2$ under the action of the known force of interest, $\delta(t)$. In particular, the discounted value at time 0 of $C$ due at time $t \geq 0$ is called its discounted present value (or, more briefly, its present value); it is equal to
$$C \exp \left[-\int_0^t \delta(s) \mathrm{ds}\right]$$ $$v(t)=\exp \left[-\int_0^t \delta(s) \mathrm{d} s\right]$$
When $t \geq 0, v(t)$ is the (discounted) present value of 1 due at time $t$. When $t<0$, the convention $\int_0^t \delta(s) \mathrm{d} s=-\int_t^0 \delta(s) \mathrm{d} s$ shows that $v(t)$ is the accumulation of 1 from time $t$ to time 0 . It follows by Eqs 2.5.2 and $2.5 .3$ that the discounted present value of $C$ due at a non-negative time $t$ is
$$C v(t)$$
In the important practical case in which $\delta(t)=\delta$ for all $t$, we may write
$$v(t)=v^t \quad \text { for all } t$$
where $v=v(1)=e^{-\delta}$. Expressions for $v$ can be easily related back to the interest rate quantities $i$ and $i^{(p)}$; this is further discussed in Chapter 3 . The values of $v^t(t=1,2,3, \ldots)$ at various interest rates are included in standard compound interest tables, including those at the end of this book.

# 金融数学代考

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|THE FORCE OF INTEREST

$$\lim {h \rightarrow 0^{+}} i_h(t)=\delta(t)$$ 符号 $h \rightarrow 0^{+}$表示限制被认为是 $h$ 趋向于“从上方“归䨌，即通过正值。当然，在趋于零的时间间隔的限制下，这总是正确的。 打电话是很正常的 $\delta(t)$ 单位时间内的兴甫力量 $t$. 鉴于方程式。 $2.4 .1, \delta(t)$ 有时称为单位时间内的名义利率 $t$ 瞬间转换。尽管它是对现实的数学理想化，但利息的力量在 复利理论中起着至关重要的作用。请注意，通过结合方程式 $2.3 .2$ 和 $2.4 .1$ ，我们可以定义 $\delta(t)$ 直接根据累积因子为 $$\delta(t)=\lim {h \rightarrow 0^{+}}\left[\frac{A(t, t+h)-1}{h}\right]$$

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|PRESENT VALUES

$$C \exp \left[-\int_{L_1}^{t_2} \hat{\delta}(t) \mathrm{d} t\right]$$

$$\begin{gathered} C \exp \left[-\int_0^t \delta(s) \mathrm{d} s\right] \ v(t)=\exp \left[-\int_0^t \delta(s) \mathrm{d} s\right] \end{gathered}$$

$C v(t)$

$v(t)=v^t \quad$ for all $t$

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## 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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