### 统计代写|金融统计代写Financial Statistics代考|AEM 4070

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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

## 统计代写|金融统计代写Financial Statistics代考|Change of time and operational time

Change of time in random processes such as $R_{s}, V_{s}$, or $N_{s}, s \geqslant 0$, refer to a transformation – tension or compression – of the timeline. This transformation may be either non-random or random; in the latter case it is usually dependent on the trajectory of the process under consideration.

The most well-known transformation is the change of time for Poisson

processes. The simplest version is a homogeneous Poisson process with constant intensity, i.e., process for which the probability of a single jump is constant per unit of time. We can make this probability equal to 1 by stretching or compressing the timeline. For such a Poisson process, called standard, the intervals between time moments of jumps are mutually independent, and the distance between any two consecutive jumps is exponentially distributed with mean 1. The realization of a homogeneous Poisson process with unit intensity can be likened to an alarm clock which buzzes at random at exponentially distributed times and is immediately reset after it.

Given a homogeneous, but not standard, Poisson process $N_{t}$ with the intensity different from unit, but for which the probability of a jump in unit time is equal to $r>0$, we can transform it into a standard process by re-scaling the time axis by $r$, i.e., stretching out the separation between jump points if the process has high intensity, or compressing it if the process has low intensity ${ }^{62}$. Writing this as a formula, we define a new time by setting $\Upsilon_{t}=r t$. The process $N_{r_{t}}$ looks like a realization of the standard Poisson process, even though $N_{t}$ does not.

If the Poisson process is inhomogeneous and its intensity $r_{t}$ depends on time, we can use a similar approach: when the intensity is high, we stretch the original timeline in order to slow down the flow of time. On the contrary, when the intensity is low, we compress the original timeline in order to speed up the flow of time. It is written as
$$r_{t}=\int_{0}^{t} r_{s} d s .$$
If $t_{1}, t_{2}, \ldots$ denote the (random) moments of jumps, then $\Upsilon_{t_{2}}, \Upsilon_{t_{2}}, \ldots$ are the transformed moments of jumps. They correspond to the moments of jumps of the standard Poisson process, and the random variables $\Upsilon_{t_{i+1}}-\Upsilon_{t_{i}}$ are independent and all exponentially distributed with mean 1 . This method allows us to reduce the study to the previous case, in which the intensity is constant and the process is a standard Poisson process.

## 统计代写|金融统计代写Financial Statistics代考|Purpose and origin of Lundberg’s model

Lundberg’s model was created during a time when only individual, essentially static models existed. These dealt with a fixed portfolio and referred to a fixed time in the future. The emergence of Lundberg’s theory was due to the rapid growth of the insurance business, which covered wider fields of applications. To regulate it, a comprehensive model of the insurance company, rather than an individual portfolio, was needed. Lundberg’s model provides a bird’s eye view of the insurance process.

In his memoir [43], entitled “Half a century with probability theory”, H. Cramér gave a brief description of the advancements made in Lundberg’s model, as follows. The net result of the risk business of an insurance company for a period of, say, one year, was considered as the sum of the results for each of the individual policies in this company’s portfolio. Assuming that the risks under all these policies are independent in the aggregate, their connection with the classical central limit theorem of probability theory was made apparent. But the entire insurance process can be considered as an economic system that develops over time and is subject to random fluctuations at each moment of time. Such systems were

considered in those pioneering works which appear as forerunners of the modern theory of stochastic processes today.

According to K. Borch (see [26], p. 439), the approach proposed by F. Lundberg exempts us from the need to consider each individual contract in the portfolio when the probability distribution of the total amount of claim payments is sought. In Lundberg’s approach, it is formed from two distributions, and both of them can be estimated using the business records of the company. From this viewpoint, it is natural to call Lundberg’s theory the collective theory of risk. However, the true innovation of this theory is its dynamic character, which distinguishes this risk theory from the former static risk theory.

## 统计代写|金融统计代写Financial Statistics代考|Main drawbacks of Lundberg’s model

The most important drawbacks of Lundberg’s model were listed by $\mathrm{K}$. Borch (see [26], p. 450 ) back in 1967 . He wrote that the following assumptions of this model are far-removed from reality:
(i) The stationarity assumption ${ }^{72}$, from which it follows that the nature of the company’s business will never change. This assumption becomes a little more reasonable than it seems at first glance, if we consider operational time.
(ii) The assumption that the probability laws underlying the whole process are fully known.
(iii) The implicit assumption that once made, a decision by management can never be changed.

None of these three items compromises the fundamental value of Lundberg’s approach in which the main idea is to consider the insurance process as the implementation of a large-scale random phenomenon. K. Borch only pointed out the weaknesses, “holes” of the model among which is the assumption of homogeneity and immutability of the probability laws throughout the entire lifetime of the company. It is easy to see that exactly these assumptions of Lundberg’s model contradict to established practice of insurance regulation and management.

H. Cramér noted that problems of this sort have been quite obvious to F. Lundberg, who was concerned about the adequacy of his model. In the historical review [41] of Lundberg’s works, he wrote that
in view of certain misconceptions that have appeared it is, however, necessary to point out that Lundberg repeatedly emphasizes the practical importance of some arrangement which automatically prevents the risk reserve from growing unduly. This point is, in fact, extensively discussed in the papers of 1909,1919 and 1926-28. One possible arrangement proposed to this end is to work with a security factor $\tau=\tau(x)$ which is a decreasing function of the risk reserve $R(t)=x$. Another possibility is to dispose, at predetermined epochs, of part of the risk reserve for bonus distribution. By either method, the growth of the risk reserve may be efficiently controlled. What Lundberg does in this connection is really to work with a rather refined case of what has much later come to be known as a random walk with two barriers.

From certain quarters, the Lundberg’s theory has been declared to be unrealistic because, it is asserted, no limit is imposed on the growth of the risk reserve. In view of what has been said above, it would seem that these critics have not read the author they are criticizing. For a non-Scandinavian author there is, of course, the excuse that most of Lundberg’s works are written in Swedish.

r吨=∫0吨rsds.

## 统计代写|金融统计代写Financial Statistics代考|Purpose and origin of Lundberg’s model

Lundberg 的模型是在只有个别的、基本上是静态的模型存在的时期创建的。这些涉及固定的投资组合，并提到未来的固定时间。伦德伯格理论的出现，是由于保险业务的快速增长，其应用领域覆盖更广。为了对其进行监管，需要一个全面的保险公司模型，而不是单个的投资组合。Lundberg 的模型提供了保险过程的鸟瞰图。

## 统计代写|金融统计代写Financial Statistics代考|Main drawbacks of Lundberg’s model

Lundberg 模型最重要的缺点如下：ķ. Borch（参见 [26]，第 450 页）早在 1967 年。他写道，该模型的以下假设与现实相去甚远：
（i）平稳性假设72，由此可见，公司的业务性质永远不会改变。如果我们考虑操作时间，这个假设变得比乍看之下更合理一些。
(ii) 假设整个过程的概率规律是完全已知的。
(iii) 隐含的假设，即一旦做出，管理层的决定永远不会改变。

H. Cramér 指出，这类问题对 F. Lundberg 来说非常明显，他担心他的模型是否充分。在对伦德伯格著作的历史回顾 [41] 中，他写道，

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