### 统计代写|金融统计代写Financial Statistics代考|ST326

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• Foundations of Data Science 数据科学基础

## 统计代写|金融统计代写Financial Statistics代考|Ways to overcome the identified problems

Insights into the strengths and weaknesses of Lundberg’s model stimulated a search for new lines of development. Two of them have branched out. First, numerous articles and monographs (see, e.g., $[167],[66],[7],[93]$ ) have published developments to weaken the technical assumptions of this model, addressing more complex probability mechanisms of insurance and using more sophisticated stochastic processes to model the risk reserve’s dynamics. Second is an attempt to reconsider the structural assumptions of Lundberg’s model by means of control theory. According to C. Philipson (see [156], p. 68), both these lines of development are based on the fundamental conception of the collective risk theory, which was created by Filip Lundberg.

The boundary between these two approaches is fuzzy. For example, the random walk model with two levels mentioned by H. Cramér in [41] is a sophistication of the traditional model of risk reserve. Therefore, it relates to the first branch of development mentioned previously. But the same model can be attributed to the second branch of development, since the upper barrier is a tool which automatically prevents the risk reserve from growing unduly.

Sharing the opinion of $\mathrm{K}$. Borch about the shortcomings of the theory of collective risks, as listed in points (i)-(iii) in Section 1.5.3.3, many experts believed the merger of the fundamental concept of Lundberg’s model with the methods of control theory is urgently needed in order to make it consistent with realities of the insurance business. The most important step was deemed by experts to be an introduction to the model of the possibility to change decisions once made.

For example, C. Philipson wrote (see [156], p. 59) that the risk premiums on which the tariff is built, called the applied risk premiums, themselves form random processes with discontinuous time parameters. Their trajectories are step functions in which each new step starts at a moment of change in the tariff rates. Additionally, if the security loading in the premiums is a product of the prediction of some measure of the variation of the risk premium, then we have a superposition of processes of a similar structure.

## 统计代写|金融统计代写Financial Statistics代考|Diffusion risk model: a useful auxiliary tool

Despite the intuitive transparency of Lundberg’s model, its results are not at all obvious. Thus, even in the simplest case of absence of migration of insureds and exponentially distributed interclaim intervals, when the claim arrival process is a homogeneous Poisson process, and when claim amounts are exponentially distributed, the expression for the probability of ruin within time $t$ written out in equality (C.3), is quite complex.

Looking at the trajectory of the risk reserve process from a considerable dis-

tance, we would see something similar to the trajectory of a Brownian motion with trend (see later shown Fig. 1.4). This suggests that the diffusion process can be used as a simplified model of the risk reserve process of an insurance company. At time $s \geqslant 0$, this model has the form
$$R_{s}=u+c s-V_{s}, \quad V_{s}=\vartheta s+\sigma(\vartheta) \mathrm{W}{s+}$$ where, as before, $u$ denotes the initial capital, $c$ is the premium intensity otherwise simply known as price. The aggregate claim payments $V{s}, s \geqslant 0$, where $\mathrm{W}{s}, s \geqslant 0$, denotes a standard Wiener process, is a diffusion process starting at the origin and having drift parameter $\vartheta>0$ and diffusion parameter $\sigma(\vartheta)>0$. It is easy to see that $E V{s}=\vartheta s$ and $\mathrm{D} V_{s}=\sigma^{2}(\vartheta) s$. If the premium intensity $c$ is calculated according to the expected value principle ${ }^{76}$, i.e., if it is taken such that the mean aggregate claim payout and aggregate premiums collected are equal to each other, it equals $\vartheta$. If $c=\vartheta(1+\tau)$, then $\tau$ is called the premium loading.

Diffusion models are commonly used in risk theory for the following reasons. Firstly, when a more accurate description of the claim payout process is not required, these models are productive (see, e.g., [8], [192]). Secondly, if we approximate the jumping process of claim payments by a corresponding diffusion process, we may obtain useful approximation results for the original model (see, e.g., $[88],[78],[85],[183],[168],[73]$, and the survey work [6]). A number of useful results for the diffusion model (1.7) is gathered in Section C.1.

To explore diffusion model $(1.7)$ by analytical methods is easier, than Lundberg’s model, but when doing so, in addition to careful attention to the purely formal inconsistencies, such as the possibility of negative insurance payments ${ }^{77}$, a warning of restricted applicability, or non-applicability, should be clearly given.

## 统计代写|金融统计代写Financial Statistics代考|Program for building a model of long-term controlled insurance

In order to preserve the main advantages of Lundberg’s model and eliminate its main drawbacks described in Section 1.5.3, and in order to bring risk theory closer to the practical needs of insurance, we must move to a model of a multi-year controlled insurance process. This model is desirable as it reflects aspects such as reinsurance, investments, dividends, and bonuses.

Finnish and British Solvency Working Parties devised (see [53], Equation (1.1.1)) the following equation
$$R^{[k \mid}=R^{|k-1|}+I^{[k]}+C^{|k|}+V_{\mathrm{re}}^{[k]}+A_{\text {new }}^{[k]}+B_{\text {new }}^{[k]}-V^{[k \mid}-E^{|k|}-I_{\mathrm{re}}^{|k|}-D^{|k|},$$
as a year-to-year transition equation describing the dynamics of an insurance company from year to year. Here $k(k=1,2, \ldots)$ is the effective period’s, or account year’s, or simply year’s number, $R^{(k \mid}$ is the amount of assets at the end, and $R^{[k-1 \mid}$ at the beginning, of the $k$-th year, $I^{k \mid}$ is the premium income in the $k$-th year, $C^{|k|}$ is the return received in respect of the investments during the $k$-th year, $V_{r e}^{|k|}$ is the recovery from reinsurers during that period, $A_{\text {new }}^{\text {te }}$ is the new equity capital issued and subscribed for during that period, $B_{\text {new }}^{[k \mid}$ is the new debt capital issued and subscribed for during the $k$-th year and any other borrowing, $V^{|k|}$ is the amount of claim payments made during the $k$-th year including payments made on account, $E^{[k]}$ is the amount of commission paid and administration and operation expenses in the $k$-th year, $I_{\mathrm{re}}^{k]}$ is ceded reinsurance premium in the $k$-th year, and $D^{[k]}$ is the dividends paid to shareholders and bonuses paid to policyholders in the $k$-th year. Overall, all the variables, such as $R^{k \mid}$, $I^{|k|}$, etc., relate to the end of the $k$-th year.
Transition equation (1.8) describes the difference between all inflows, or revenues, and all outflows, or expenditures. Therefore, by its very nature, it is similar to equality (1.2). The obvious difference between the two of them is that the transition equation (1.8) reckons periods (insurance years), as it is usually done in the profit and loss accounting; traditional practice is that each accounting period ends by summarizing and making reports, and starts with developing administrative and control decisions ${ }^{78}$ on this basis.

## 统计代写|金融统计代写Financial Statistics代考|Diffusion risk model: a useful auxiliary tool

tance，我们会看到类似于带有趋势的布朗运动的轨迹（见后面的图 1.4）。这表明扩散过程可以作为保险公司风险准备金过程的简化模型。当时s⩾0, 这个模型有形式

Rs=在+Cs−在s,在s=ϑs+σ(ϑ)在s+和以前一样，在表示初始资本，C是溢价强度，也简称为价格。总索赔付款在s,s⩾0， 在哪里在s,s⩾0，表示标准维纳过程，是从原点开始并具有漂移参数的扩散过程ϑ>0和扩散参数σ(ϑ)>0. 很容易看出和在s=ϑs和D在s=σ2(ϑ)s. 如果溢价强度C是按照期望值原理计算的76，即，如果平均总索赔支出和收取的总保费彼此相等，则等于ϑ. 如果C=ϑ(1+τ)， 然后τ称为溢价加载。

## 统计代写|金融统计代写Financial Statistics代考|Program for building a model of long-term controlled insurance

R[ķ∣=R|ķ−1|+我[ķ]+C|ķ|+在r和[ķ]+一个新的 [ķ]+乙新的 [ķ]−在[ķ∣−和|ķ|−我r和|ķ|−D|ķ|,

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