## 数学代写|离散数学作业代写discrete mathematics代考|CS3653

2023年1月3日

couryes-lab™ 为您的留学生涯保驾护航 在代写离散数学discrete mathematics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。我们的专家在代写离散数学discrete mathematics代写方面经验极为丰富，各种代写离散数学discrete mathematics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
couryes™为您提供可以保分的包课服务

## 数学代写|离散数学作业代写discrete mathematics代考|The Gambler’s ruin

To conclude this chapter and as an introduction to problems in financial mathematics, let us look at a problem of ruin. A gambler has an initial capital $x$.

They toss for heads or tails using a balanced coin and win 1 if they obtain tails and loses 1 if they obtain heads. The gambler has a fixed objective of a fortune $a \geq x$ and a baseline for losing, $b \leq x$. They play until their fortune reaches $a$ or $b$.

The gambler’s fortune is modeled using a random walk $\left(S_n\right){n \in \mathbb{N}}$, with $S_0=x$ and $S_n=S_0+\sum{k=1}^n X_k$, where $X_k$ represents the winnings of the $k$-th toss.

It is known that arriving at $a$ or $b$, starting from $x$ is equivalent to arriving at $a-x$ or $b-x$ starting from 0 . The game then stops almost surely at the end of a finite time.
We use $p_x$ to denote the probability of ruin, starting from an initial capital of $x$, and $R_x$ to denote the event ruined after starting from $x$, that is,
$$p_x=\mathbb{P}\left(T_b<T_a\right)=\mathbb{P}\left(R_x\right) .$$
It can be directly noted that $p_a=0$ and $p_b=1$ because in both these situations the game does not start; ruin is impossible in the first case and is certain in the second.
We will now obtain a recurrence over $p_x$. If $b<x<a$, then using the formula for total probabilities, we have
\begin{aligned} p_x & =\mathbb{P}\left(R_x\right) \ & =\mathbb{P}\left(R_x, \mid S_1=x+1\right) \mathbb{P}\left(S_1=x+1\right)+\mathbb{P}\left(R_x \mid S_1=x-1\right) \mathbb{P}\left(S_1=x-1\right) \ & =\mathbb{P}\left(R_{x+1}\right) \frac{1}{2}+\mathbb{P}\left(R_{x-1}\right) \frac{1}{2}, \end{aligned}
by using stationarity.

## 数学代写|离散数学作业代写discrete mathematics代考|Martingale transform

THEOREM 4.1.-Let $\left(K_n\right){n \in \mathbb{N}}$ be a positive and $\left(\mathcal{F}_n\right){n \in \mathbb{N}}$ be a predictable process. Consider $\left(X_n\right){n \in \mathbb{N}}$ an $\left(\mathcal{F}_n\right){n \in \mathbb{N}}$-martingale [respectively submartingale, supermartingale ]. If the process $\left(K_n\right){n \in \mathbb{N}}$ is bounded, then the process $\left(K \cdot X_n\right){n \in \mathbb{N}}$, defined by $K \cdot X_0=X_0$ and for any $n \geq 1$,
$$K \cdot X_n:=\sum_{k=1}^n K_k\left(X_k-X_{k-1}\right),$$
is an $\left(\mathcal{F}n\right){n \in \mathbb{N}}-$ martingale [respectively submartingale, supermartingale]. The process $\left(K \cdot X_n\right){n \in \mathbb{N}}$ is called a martingale transform of $\left(X_n\right)$. $K \cdot X_n:=\sum{k=1}^n K_k\left(X_k-X_{k-1}\right)$, is clearly $\mathcal{F}n$-measurable. On the contrary, because $\left(K_n\right)$ is bounded, there exists $M>0$ such that $$\mathbb{E}\left[\left|K \cdot X_n\right|\right] \leq M \sum{k=1}^n \mathbb{E}\left[\left|X_k-X_{k-1}\right|\right]<\infty,$$
because $\left(X_n\right)$ is an integrable.

Finally, for any $n \geq 1$, upon simplification we obtain:
\begin{aligned} \mathbb{E}\left[K \cdot X_n-K \cdot X_{n-1} \mid \mathcal{F}{n-1}\right] & =\mathbb{E}\left[K_n\left(X_n-X{n-1}\right) \mid \mathcal{F}{n-1}\right] \ & -K_n \mathbb{E}\left[X_n-X{n-1} \mid \mathcal{F}{n-1}\right], \end{aligned} where the final equality is due to the fact that $\left(K_n\right)$ is $\left(\mathcal{F}_n\right){n \in \mathbb{N}}$-predictable. Now, when $\left(X_n\right)$ is an $\left(\mathcal{F}n\right){n \in \mathbb{N}}$ martingale [respectively submartingale, supermartingale], we have
$$\mathbb{E}\left[X_n-X_{n-1} \mid \mathcal{F}_{n-1}\right]=0[\text { resp. } \geq 0, \leq 0] .$$
The result of the theorem follows from this.

# 离散数学代写

## 数学代写|离散数学作业代写discrete mathematics代考|The Gambler’s ruin

$$p_x=\mathbb{P}\left(T_b<T_a\right)=\mathbb{P}\left(R_x\right) .$$

$$p_x=\mathbb{P}\left(R_x\right) \quad=\mathbb{P}\left(R_x, \mid S_1=x+1\right) \mathbb{P}\left(S_1\right.$$

## 数学代写|离散数学作业代写discrete mathematics代考|Martingale transform

$K \cdot X_n:=\sum k=1^n K_k\left(X_k-X_{k-1}\right)$, 显然 $\mathcal{F} n$ 可衡量的。相反，因为 $\left(K_n\right)$ 是有界的，存在 $M>0$ 这 样
$$\mathbb{E}\left[\left|K \cdot X_n\right|\right] \leq M \sum k=1^n \mathbb{E}\left[\left|X_k-X_{k-1}\right|\right]<\infty$$

$$\mathbb{E}\left[K \cdot X_n-K \cdot X_{n-1} \mid \mathcal{F} n-1\right]=\mathbb{E}\left[K _ { n } \left(X_n-X n\right.\right.$$

## 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。