# 金融代写|风险理论代写Risk theory代考|STAT4901

## 金融代写|风险理论代写Risk theory代考|Tails of Sums of Light-Tailed Random Variables

We consider here the asymptotics of $\overline{F^{* n}}(x)=\mathbb{P}\left(S_n>x\right)$ and the corresponding density $f^{* n}(x)$ for an i.i.d. light-tailed sum as $x$ goes to $\infty$ with $n$ fixed. This is similar to the set-up for the subexponential limit theory developed in Sect. 2, but differs from the large deviations set-up of Sect. 3 where $n$ and $x$ went to $\infty$ at the same time. The theory which has been developed applies to distributions with a density or tail close to $\mathrm{e}^{-c x^\beta}$ where $\beta \geq 1$.

The cases $\beta=1$ and $\beta>1$ are intrinsically different (of course, $\beta<1$ leads to a heavy tail). The difference may be understood by looking at two specific examples from Sect. 1 with $n=2$. The first is the exponential distribution in Example $1.3$ where $\beta=1$, which shows that if $X_1+X_2>x$, then approximately $X_1, X_2$ are both uniform between 0 and $x$, with the joint distribution concentrating on the line $x_1+$ $x_2=x$. On the other hand, for the normal distribution where $\beta=2$, Example $1.4$ shows that $X_1, X_2$ are both approximately equal to $x / 2$ if $X_1+X_2>x$.

For ease of exposition, we make some assumptions that are simplifying but not all crucial. In particular, we take $F$ to be concentrated on $(0, \infty)$ and having a density $f(x)$, and exemplify ‘close to’ by allowing a modifying regularly varying prefactor to $\mathrm{e}^{-c x^\beta}$. For the density, this means
$$f(x) \sim L(x) x^{\alpha+\beta-1} \mathrm{e}^{-c x^\beta}, \quad x>0,$$
with $L(\cdot)$ slowly varying. However, in the rigorous proofs we only take $n=2$ and $L(x) \equiv d$ will be constant. We shall need the following simple lemma (cf. Exercise 4.1):

Lemma 4.1 If (4.1) holds with $\beta \geq 1$, then $\bar{F}(x)=\mathbb{P}(X>x) \sim$ $L(x) x^\alpha \mathrm{e}^{-c x^\beta} / c \beta$

## 金融代写|风险理论代写Risk theory代考|Aggregate Claims and Compound Sums: Generalities

Let $A$ be the total amount of claims to be covered by an insurance company in a given time period (say one year). The company will possess statistics to estimate basic quantities like $\mu_A=\mathbb{E} A$ and will have made sure that the premiums $p$ received during the same time period exceed $\mu_A$ with some margin covering administration costs, profits to shareholders, etc. Nevertheless, due to the random nature of $A$, one is faced with an uncertainty as to whether in fact $A$ is much larger than $\mu_A$, and what is the probability that such unfortunate events occur. The topic of this and the following section is to present methods for quantitative evaluations of risks of this type.

Risk evaluation, as for any kind of probability calculation, requires a model. We will assume that $A=V_1+\cdots+V_N$, where $N$ is the total number of claims, $V_n$ is the size of the $n$th claim, and that the $V_n$ are i.i.d. with common distribution $F$ and independent of $N$, an $\mathbb{N}$-valued r.v. with point probabilities $p_n=\mathbb{P}(N=n)$. The problem is to say something about $\mathbb{P}(A>x)$ for large $x$ under these assumptions. Often the purpose of this is VaR (Value-at-Risk) calculations, that is, the evaluation of $\alpha$-quantiles $q_\alpha(A)$, say for $\alpha=95,97.5$ or $99 \%$.

We will start with some elementary analytical formulas for transforms, moments, etc. For notation and terminology, see Sect. A.11 in the Appendix.

Proposition $5.1$ The m.g.f. of $A$ is $\widehat{p}[\widehat{F}[\theta]]$, where $\widehat{p}[z]=\mathbb{E} z^N$ is the p.g.f. of $N$ and $\widehat{F}[\theta]=\mathbb{E} e^{\theta V}$ the m.g.f. of F. Further,
$$\mu_A=\mu_N \mu_V, \sigma_A^2=\sigma_N^2 \mu_V^2+\mu_N \sigma_V^2,$$
where $\mu_A=\mathbb{E} A, \sigma_V^2=\mathbb{V a r} V$, etc.

## 金融代写|风险理论代写Risk theory代考|Tails of Sums of Light-Tailed Random Variables

$$f(x) \sim L(x) x^{\alpha+\beta-1} \mathrm{e}^{-c x^3}, \quad x>0,$$

## 金融代写|风险理论代写Risk theory代考|Aggregate Claims and Compound Sums: Generalities

$$\mu_A=\mu_N \mu_V, \sigma_A^2=\sigma_N^2 \mu_V^2+\mu_N \sigma_V^2,$$

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