### 数学代考|金融数学代考Financial Mathematics代写|Elements of Financial Mathematics: From Interest Theory to Options

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

## 数学代考|金融数学代考Financial Mathematics代写|Consumers

The population consists of overlapping generations of finitely-lived agents who are identical in every respect except for their health type. Agents live for a maximum of two periods, termed ‘youth’ (superscript $y$ ) and ‘old age’ (o). At birth each agent learns her health status as proxied by the survival probability, $\mu$. This is where the difference between health types comes in: unhealthy agents have a higher risk of dying, and therefore a shorter expected life span (which equals $1+\mu$ periods). We assume that cohorts are sufficiently large such that there is no aggregate uncertainty and probabilities and frequencies coincide. For example, the fraction of young agents of type $\mu$ who die after the first period equals exactly $1-\mu$. Note that from the perspective of an individual agent, lifetime uncertainty is resolved at the start of the second period. When still alive, the agent will live for exactly one additional period.

Labour supply is exogenous. During youth the agent is fully employed while during old age labour supply is only a fraction $\lambda$ of the unit time endowment as a result of mandatory retirement $(0<\lambda<1)$. The expected lifetime utility of a representative agent of health type $\mu$ who is born in period $t$ is given by: $$\mathbb{E} \Lambda_{t}(\mu) \equiv U\left(C_{t}^{y}(\mu)\right)+\mu \beta U\left(C_{t+1}^{o}(\mu)\right),$$ where $C_{t}^{y}(\mu)$ and $C_{t+1}^{o}(\mu)_{\text {are }}$ consumption during youth and old age, respectively, $\beta$ is a parameter capturing pure time preference $(0<\beta<1)$, and $U(\cdot)$ is the felicity function: $$U(x) \equiv \frac{x^{1-1 / \sigma}-1}{1-1 / \sigma}, \quad \sigma>0 .$$
This functional form is chosen for analytical convenience and it implies a constant intertemporal substitution elasticity, $\sigma$. We assume that the agent does not have a bequest motive such that she does not derive any utility from wealth that remains after her death.

## 数学代考|金融数学代考Financial Mathematics代写|Demography

Let $\mathrm{h}(\mu)$ denote the probability density function of health types in a given cohort upon its arrival. Then the distribution of agents in the cohort born at time $t$ can be written as:
$$L_{t}(\mu) \equiv h(\mu) L_{t},$$

$\int_{\mu_{l}}^{\mu_{h}} h(\mu) d \mu=\int_{\mu_{l}}^{\mu_{h}} d H(\mu)=1$ where $\mathrm{H}(\mu)$ is the cumulative density function. The density of $\mu$-type agents alive at time $t$ is given by $P_{t}(\mu) \equiv$ $\mu L_{t-1}(\mu)+L_{t}(\mu)$. Assuming that newborn cohorts evolve according to $L_{t}=$ $(1+n) L_{t-1}$ (with $\left.n>-1\right)$ we thus find that:
$$P_{t}(\mu)=\frac{1+\mu+n}{1+n} L_{t}(\mu) .$$
The total population alive in period $t$ is obtained by aggregating over health types:
$$P_{t} \equiv \int_{\mu_{t}}^{\mu_{\phi}} P_{t}(\mu) d \mu=\frac{1+\bar{\mu}+n}{1+n} L_{t},$$
where $\bar{\mu} \equiv \int_{\mu \mu}^{\mu_{h}} \mu h(\mu) d \mu$ is the average survival rate in the population as a whole.
Government
In the absence of annuity markets we have to make an assumption about how the accidental bequests left by the dead are redistributed among the agents who are still alive. We therefore introduce a government sector which collects the bequests and uses them to finance lump-sum income transfers $Z_{t}$ to the young. ${ }^{3}$ The government budget constraint is given by:
$$\left(1+r_{t}\right) \int_{\mu t_{t}}^{\mu_{t}}(1-\mu) L_{t-1}(\mu) S_{t-1}(\mu) d \mu=L_{t} \mathrm{Z}_{t} .$$
That is, the total amount of accidental bequests (left-hand side) equals the sum of income transfers (right-hand side).

## 数学代考|金融数学代考Financial Mathematics代写|Production

0The production side of this closed economy features a large number of perfectly competitive firms who produce a homogeneous commodity. The technology is represented by the following Cobb-Douglas production function:
$$Y_{t}=\Omega_{0} K_{t}^{\varepsilon} N_{t}^{1-\varepsilon}, \quad 0<\varepsilon<1,$$ where $Y_{t}$ is total output, $\Omega_{0}>0$ is an exogenously given index of general factor productivity, $\mathrm{K}{\mathrm{t}}$ is the aggregate capital stock, and $N{t} \equiv L_{t}+\bar{\mu} \lambda L_{t-1}$ is the labour force. By defining $y_{t} \equiv Y_{t} / N_{t}$ and $k_{t} \equiv K_{t} / N_{t}$ we can write the intensive-form production function as:
$$y_{t}=\Omega_{0} k_{t}^{\varepsilon} .$$
Profit-maximizing behaviour of firms yields the following factor demand equations:
\begin{aligned} r_{t}+\delta &=\varepsilon \Omega_{0} k_{t}^{\varepsilon-1}, \ w_{t} &=(1-\varepsilon) \Omega_{0} k_{t}^{\varepsilon}, \end{aligned}
where $\delta$ is the constant rate of depreciation of the capital stock $(0<\delta<1)$. The general model without annuities is fully characterized by the following fundamental difference equation:
$$k_{t+1}=\phi_{1}^{T Y}\left(r_{t+1}\right)\left[w_{t}+Z_{t}\right]-\phi_{2}^{T Y}\left(r_{t+1}\right) \frac{\lambda w_{t+1}}{1+r_{t+1}},$$
where $\phi_{1}^{T \gamma}\left(r_{t+1}\right)$ and $\phi_{2}^{T \gamma}\left(r_{i+1}\right)$ are given by:
\begin{aligned} &\phi_{1}^{T \gamma}\left(r_{t+1}\right) \equiv \frac{1}{1+n+\lambda \bar{\mu}} \int_{\mu_{t}, \nu}^{\mu_{t}}\left[1-\Phi\left(\mu, 1+r_{t+1}\right)\right] h(\mu) d \mu . \ &\phi_{2}^{T \gamma}\left(r_{t+1}\right) \equiv \frac{1}{1+n+\lambda \bar{\mu}} \int_{\mu_{k i t}}^{\mu_{\mu}} \Phi\left(\mu, 1+r_{t+1}\right) h(\mu) d \mu . \end{aligned}
Equation (19) is obtained by imposing equilibrium in the savings market and using the cohort size evolutions described in Sect. 2.2. At time $t$ the predetermined capital intensity, $\mathrm{k}{\mathrm{t}}$, pins down $\mathrm{r}{\mathrm{t}}, \mathrm{w}{\mathrm{t}}$, and $\mathrm{Z}{\mathrm{t}}$, so that (19) in combination with (17) and (18) constitutes an implicit function determining $\mathrm{k}{\mathrm{t}+1}, \mathrm{r}{\mathrm{t}+1}$, and $\mathrm{w}_{\mathrm{t}+1}$.

## 数学代考|金融数学代考Financial Mathematics代写|Demography

∫μlμHH(μ)dμ=∫μlμHdH(μ)=1在哪里H(μ)是累积密度函数。的密度μ型特工当时还活着吨是（谁）给的磷吨(μ)≡ μ大号吨−1(μ)+大号吨(μ). 假设新生儿队列根据大号吨= (1+n)大号吨−1（和n>−1)因此，我们发现：

(1+r吨)∫μ吨吨μ吨(1−μ)大号吨−1(μ)小号吨−1(μ)dμ=大号吨从吨.

## 数学代考|金融数学代考Financial Mathematics代写|Production

0这种封闭经济的生产方面有大量完全竞争的公司生产同质商品。该技术由以下 Cobb-Douglas 生产函数表示：

r吨+d=eΩ0ķ吨e−1, 在吨=(1−e)Ω0ķ吨e,

ķ吨+1=φ1吨是(r吨+1)[在吨+从吨]−φ2吨是(r吨+1)λ在吨+11+r吨+1,

φ1吨C(r吨+1)≡11+n+λμ¯∫μ吨,νμ吨[1−披(μ,1+r吨+1)]H(μ)dμ. φ2吨C(r吨+1)≡11+n+λμ¯∫μķ一世吨μμ披(μ,1+r吨+1)H(μ)dμ.

## 有限元方法代写

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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。