# 金融代写|风险理论代写Risk theory代考|MATH4128

## 金融代写|风险理论代写Risk theory代考|Poisson Compounding

In Chap. IV, we will meet geometric compounding, i.e. $p_n=(1-\rho) \rho^n$, as a technical tool in ruin theory. However, by far the most important case from the point of view of aggregate claims is Poisson compounding, $p_n=\mathrm{e}^{-\lambda} \lambda^n / n !$. As explained in Sect. I.5, the Poisson assumption can be motivated in a number of different, though often related, ways. For example:

1. Assume that the number of policy holders is $M$, and that the $m$ th produces claims according to a Poisson process with rate (intensity) $\lambda_m$. In view of the interpretation of the Poisson process as a model for events which occur in an ‘unpredictable’ and time-homogeneous way, this is a reasonable assumption for example in car insurance or other types of accident insurance. If further the policy holders behave independently, then the total claims process (the superposition of the individual claim processes) is again Poisson, now with rate $\lambda=\lambda_1+\cdots+\lambda_M$. In particular, if $N$ is the number of claims in a time period of unit length, then $N$ has a Poisson $(\lambda)$ distribution.
2. Assume again that the number of policy holders is $M$, and that the number of claims produced by each during the given time period is 0 or 1, w.p. $p$ for 1 . If $p$ is small and $M$ is large, the law of small numbers (Theorem 5.1) therefore implies that $N$ has an approximate Poisson( $\lambda$ ) distribution where $\lambda=M p$.

In the Poisson case, $\widehat{p}[z]=\mathrm{e}^{\lambda(z-1)}$, and therefore the c.g.f. of $A$ is (replace $z$ by $\widehat{F}[\theta]$, cf. Proposition I.5.1, and take logarithms)
$$\kappa_A(\theta)=\lambda(\widehat{F}[\theta]-1) .$$
Further, Proposition $5.1$ yields
$$\mu_A=\lambda \mu_V, \quad \sigma_A^2=\lambda \sigma_V^2+\lambda \mu_V^2=\lambda \mathbb{E} V^2$$

## 金融代写|风险理论代写Risk theory代考|The Saddlepoint Approximation

We consider the case of a Poisson $N$ where $\mathbb{E e}^{\alpha A}=\mathrm{e}^{\kappa(\alpha)}$ with $\kappa(\alpha)=\lambda(\widehat{F}[\alpha]-1)$. The exponential family generated by $A$ is given by
$$\mathbb{P}\theta(A \in \mathrm{d} x)=\mathbb{E}\left[\mathrm{e}^{\theta A-\kappa(\theta)} ; A \in \mathrm{d} x\right] .$$ In particular, $$\kappa\theta(\alpha)=\log \mathbb{E}\theta \mathrm{e}^{\alpha A}=\kappa(\alpha+\theta)-\kappa(\theta)=\lambda\theta\left(\widehat{F}\theta[\alpha]-1\right),$$ where $\lambda\theta=\lambda \widehat{F}[\theta]$ and $F_\theta$ is the distribution given by
$$F_\theta(\mathrm{d} x)-\frac{\mathrm{e}^{\theta x}}{\widehat{F}[\theta]} F(\mathrm{~d} x) .$$
This shows that the $\mathbb{P}\theta$-distribution of $A$ has a similar compound Poisson form as the $\mathbb{P}$-distribution, only with $\lambda$ replaced by $\lambda\theta$ and $F$ by $F_\theta$.

Following the lines of Sect. 3.4, we shall derive a tail approximation by exponential tilting. For a given $x$, we define the saddlepoint $\theta=\theta(x)$ by $\mathbb{E}\theta A=x$, i.e. $\kappa\theta^{\prime}(0)=\kappa^{\prime}(\theta)=x$.
Proposition 7.3 Assume that $\lim {r \uparrow r^} \widehat{F}^{\prime \prime}[r]=\infty$, $$\lim {r \uparrow r^} \frac{\widehat{F}^{\prime \prime \prime}[r]}{\left(\widehat{F}^{\prime \prime}[r]\right)^{3 / 2}}=0,$$
where $r^*=\sup {r: \widehat{F}[r]<\infty}$. Then as $x \rightarrow \infty$, $$\mathbb{P}(A>x) \sim \frac{\mathrm{e}^{-\theta x+\kappa(\theta)}}{\theta \sqrt{2 \pi \lambda \widehat{F}^{\prime \prime}[\theta]}} .$$

## 金融代写|风险理论代写Risk theory代考|Poisson Compounding

1. 假设保单持有人的数量为 $M$ ，并且那个 $m$ th 根据具有速率 (强度) 的 Poisson 过程产生声明 $\lambda_m$. 鉴于 将泊松过程解释为以“不可预测”和时间均匀的方式发生的事件的模型，这是一个合理的假设，例如在汽 车保险或其他类型的事故保险中。如果保单持有人进一步独立行事，那么总索赔过程 (个人索赔过程 的喗加）再次是泊松，现在有利率 $\lambda=\lambda_1+\cdots+\lambda_M$. 特别是，如果 $N$ 是单位长度的时间段内的索赔 数量，那么 $N$ 有泊松 $(\lambda)$ 分配。
2. 再次假设保单持有人的数量为 $M$ ，并且每个人在给定时间段内产生的索赔数量为 0 或 $1 ， w p p$ 为 1 。 如果 $p$ 很小而且 $M$ 很大，因此小数定律（定理 5.1) 意味着 $N$ 有一个近似泊松 ( $\lambda$ ) 分布在哪里 $\lambda=M p$.
在泊松情况下， $\hat{p}[z]=\mathrm{e}^{\lambda(z-1)}$ ，因此 $\operatorname{cgf}$ 的 $A$ 是（替换 $z$ 经过 $\widehat{F}[\theta]$ ，参见。命题 I.5.1，取对数)
$$\kappa_A(\theta)=\lambda(\widehat{F}[\theta]-1) .$$
此外，命题5.1产量
$$\mu_A=\lambda \mu_V, \quad \sigma_A^2=\lambda \sigma_V^2+\lambda \mu_V^2=\lambda \mathbb{E} V^2$$

## 金融代写|风险理论代写Risk theory代考|The Saddlepoint Approximation

$$\mathbb{P} \theta(A \in \mathrm{d} x)=\mathbb{E}\left[\mathrm{e}^{\theta A-\kappa(\theta)} ; A \in \mathrm{d} x\right] .$$

$$\kappa \theta(\alpha)=\log \mathbb{E} \theta \mathrm{e}^{\alpha A}=\kappa(\alpha+\theta)-\kappa(\theta)=\lambda \theta(\widehat{F} \theta[\alpha]-1),$$

$$F_\theta(\mathrm{d} x)-\frac{\mathrm{e}^{\theta x}}{\widehat{F}[\theta]} F(\mathrm{~d} x) .$$

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部:

myassignments-help服务请添加我们官网的客服或者微信/QQ，我们的服务覆盖：Assignment代写、Business商科代写、CS代考、Economics经济学代写、Essay代写、Finance金融代写、Math数学代写、report代写、R语言代考、Statistics统计学代写、物理代考、作业代写、加拿大代考、加拿大统计代写、北美代写、北美作业代写、北美统计代考、商科Essay代写、商科代考、数学代考、数学代写、数学作业代写、physics作业代写、物理代写、数据分析代写、新西兰代写、澳洲Essay代写、澳洲代写、澳洲作业代写、澳洲统计代写、澳洲金融代写、留学生课业指导、经济代写、统计代写、统计作业代写、美国Essay代写、美国代考、美国数学代写、美国统计代写、英国Essay代写、英国代考、英国作业代写、英国数学代写、英国统计代写、英国金融代写、论文代写、金融代考、金融作业代写。

Scroll to Top