## 二维回归相依风险模型的精细大偏差

1 浙江工商大学统计与数学学院 杭州 310018

2 浙大城市学院计算机与计算科学学院 杭州 310015

## Precise Large Deviations for a Bidimensional Risk Model with the Regression Dependent Structure

Chen Zhenlong,1, Liu Yang,1, Fu Ke-ang,1,2

1 School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018

2 Department of Statistics, Zhejiang University City College, Hangzhou 310015

 基金资助: 国家自然科学基金.  11971432国家社会科学基金.  20BTJ050浙江省自然科学基金.  LY21G010003浙江省重点建设高校优势特色学科（浙江工商大学统计学）

 Fund supported: the NSFC.  11971432the NSSFC.  20BTJ050the NSF of Zhejiang Province.  LY21G010003the Zhejiang Province Focuses on the Construction of Advantageous and Characteristic Disciplines in Universities(Statistics of Zhejiang Industrial and Commercial University)

Abstract

In this paper, we consider a non-standard bidimensional risk model, in which the claim sizes $\{\vec{X}_k=(X_{1k}, X_{2k})^T,$ $k\ge 1\}$ form a sequence of independent and identically distributed random vectors with nonnegative components that are allowed to be dependent on each other, and there exists a regression dependent structure between these vectors and the inter-arrival times. By assuming that the univariate marginal distributions of claim vectors have consistently varying tails, we obtain the precise large deviation formulas for the bidimensional risk model with the regression dependent structure.

Keywords： Bidimensional risk model ; Consistently varying tail ; Precise large deviations ; Regression dependence

Chen Zhenlong, Liu Yang, Fu Ke-ang. Precise Large Deviations for a Bidimensional Risk Model with the Regression Dependent Structure. Acta Mathematica Scientia[J], 2022, 42(3): 934-942 doi:

## 1 引言

$$$\vec{S}(t)=\sum\limits_{k=1}^{N(t)}{\vec{X}_k}, \; \; \; t\ge 0.$$$

$\{\theta_k, k\ge 1\} $$\{ \vec{X}_{k}, k\ge1\} 相互独立, 且 \{\theta_k, k\ge 1\} 为独立同分布的随机变量序列, 那么模型(1.1)就是标准的二维更新模型, 在经典风险理论的研究中起到了重要作用. 我们知道在现实中(不妨以车辆损失险为例), 保险公司往往会规定若索赔次数增多将导致保费折扣降低, 于是很多小额索赔将不会报告给保险公司, 这就使得索赔发生的间隔时间被延长了. 因此, 在保险实务中, 索赔额与索赔发生的时间间隔(或索赔计数过程)之间往往是具有一定相依关系的. 随着保险业经营日趋复杂, 文献[4-6]中关于索赔额与索赔发生时间间隔的独立性假设就显得不现实了. 为解决这一问题, 众多学者提出了多种放宽索赔额与索赔发生时间间隔独立性假设的非标准更新风险模型[7-10]. 其中, Bi和Zhang[9]首次基于网络马氏骨架过程的思想, 提出了半马尔可夫型相依结构, 具体表达式如下 其中 \{Y_k, k\ge1\} 是独立同分布的一维索赔额变量序列, \{\theta_k, k\ge 1\} 的定义如前文所述. 这就意味着索赔发生的等待时间不仅受此次索赔额的影响, 而且还受到前一次索赔额的影响. 随后, Li等[10]提出了回归相依结构, 即对于任意给定的 1\le d <n , 有 $${\rm P}(\theta_n>t|Y_k, k\ge 1)={\rm P}(\theta_n>t|Y_{n-d}, \cdots, Y_n).$$ 显然, 这种新的相依性结构可视为半马尔可夫型相依性结构的推广. 在保险公司的实际运营中, 索赔发生的等待时间往往与已发生的数次临近索赔额大小有关, 所以这类相依结构更符合保险实务. 受此启发, 本文将上述回归相依结构推广至二维环境下, 提出了二维回归相依结构, 并在索赔额向量内部两分量间存在一定相依关系的条件下, 得到了二维风险模型(1.1) 的精细大偏差. ## 2 预备知识及主要结论 首先引入几类常用符号, 对于两个正值函数 g(x)$$ h(x)$, 若$\limsup\limits_{x\rightarrow \infty} g(x)/ $$h(x)\le 1 , 则记作 g(x)\lesssim h(x)$$ h(x)\gtrsim g(x)$; 若$\lim\limits_{x\rightarrow \infty}g(x)/h(x)=0$, 则记作$g(x)=o(h(x)); $$\lim\limits_{x\rightarrow \infty}g(x)/h(x)=1 , 则记作 g(x)\sim h(x). 对于二元正值函数 g(x, t)$$ h(x, t)$, 若$\limsup\limits_{t\rightarrow \infty} $$\sup\limits_{x\in \Delta _t}$$ g( x, t )/h( x, t )\le 1$, 那么我们就说当$t\rightarrow \infty$时, $g( x, t ) \lesssim h( x, t )$关于$x\in \Delta_t\ne \emptyset$一致成立.

$$$\frac{\overline{F}( x )}{\overline{F}( x\upsilon )}\le C\upsilon^p, \; x\upsilon\ge x\ge x_0.$$$

$\begin{eqnarray} {\rm P}(\theta _k>t|\vec{X}_j, j\ge 1)=\left\{\begin{array}{ll} {\rm P}(\theta _k>t|\vec{X}_{1}, \ldots, \vec{X}_{k}), & k\le d, \\ {\rm P}(\theta _k>t|\vec{X}_{k-d}, \ldots, \vec{X}_{k}), & k>d. \end{array}\right. \end{eqnarray}$

$$${\rm P}( \vec{S}_n-n\vec{\mu }>\vec{x} ) \sim n^2\overline{F}_1(x_1) \overline{F}_2( x_2), \; n\rightarrow \infty,$$$

根据文献[17]中引理3的证明方法, 稍作修改后即可知引理3.2成立.

$$${\rm P}\Big( \sum\limits_{k=1}^n{\vec{X}_k>\vec{x}, \tau_n\le t}\Big)\le Cn^{2p+2}\overline{F}_1(x_1) \overline{F}_2(x_2){\rm P}(\tau_{n-2d-1}\le t ),$$$

显然, 由$\theta_n $$\vec{X}_1, \vec{X}_2, \cdots, \vec{X}_{n-d-1},$$ \vec{X}_{n+1}, \cdots$之间的独立性可知

$\begin{eqnarray} {\rm P}\Big( \sum\limits_{k=1}^n{\vec{X}_k>\vec{x}, \tau_n\le t}\Big) &\le&{\rm P}\Big(\bigcup\limits_{i=1}^{n}(X_{1i}>{x_1}/{n}), \bigcup\limits_{j=1}^{n}(X_{2j}>{x_2}/{n}), \tau_n\le t \Big) {}\\ &\le&\sum\limits_{1\le i, j\le n}{\rm P}\big(X_{1i}>{x_1}/{n}, X_{2j}>{x_2}/{n}, \theta_1+\theta_2+\cdots+\theta_n\le t \big) \\ &\le&\sum\limits_{1\le i\le n}{\rm P}\big(X_{1i}>{x_1}/{n}, X_{2i}>{x_2}/{n}, \sum\limits_{k=1}^{n}\theta_k-\theta_i\le t \big) \\ &&+\sum\limits_{1\le i\neq j\le n}{\rm P}\big(X_{1i}>{x_1}/{n}, X_{2j}>{x_2}/{n}, \sum\limits_{k=1}^{n}\theta_k-\theta_i-\theta_j\le t \big) \\ &\le &n{\rm P}(X_{1}>{x_1}/{n}, X_{2}>{x_2}/{n}){\rm P}( \tau _{n-d-1}\le t ) \\ &&+n( n-1 ) {\rm P}( X_{1}>{x_1}/{n} ) {\rm P}(X_{2}>{x_2}/{n}){\rm P}(\tau _{n-2d-1}\le t), \end{eqnarray}$

$\begin{eqnarray} {\rm P}\Big( \sum\limits_{k=1}^n{\vec{X}_k>\vec{x}, \tau_n\le t}\Big) &\le &C\Big(n^{2p+1}\overline {F}_1(x_1)\overline {F}_2(x_2){\rm P}(\tau_{n-d-1}\le t) \\ &&+n^{2p+1}( n-1 )\overline {F}_1(x_1)\overline {F}_2(x_2){\rm P}(\tau _{n-2d-1}\le t)\Big) \\ &\le& Cn^{2p+2}\overline {F}_1(x_1)\overline {F}_2(x_2){\rm P}(\tau_{n-2d-1}\le t), \end{eqnarray}$

在下文中每个极限都看成当$t\rightarrow \infty$时, 对所有的$\vec{x}\ge \vec{\gamma }t$一致成立. 因此, 为了证明定理2.1成立, 只需证明

$\begin{eqnarray} {\rm P}(\vec{S}(t)-\vec{\mu}\lambda t>\vec{x})\lesssim (\lambda t )^2\overline{F}_1(x_1) \overline{F}_2(x_2) \end{eqnarray}$

$\begin{eqnarray} {\rm P}(\vec{S}(t)-\vec{\mu}\lambda t>\vec{x} )\gtrsim (\lambda t)^2\overline{F}_1(x_1) \overline{F}_2(x_2) \end{eqnarray}$

$\begin{eqnarray} {\rm P}\big(\vec{S}(t)-\vec{\mu }\lambda t>\vec{x}\big)&=&{\rm P}\big(\vec{S}(t)-\vec{\mu }\lambda t>\vec{x}, N(t)\le\lambda t+\varepsilon t\big) \\ &&+{\rm P}\big(\vec{S}(t)-\vec{\mu}\lambda t>\vec{x}, N(t)>\lambda t+\varepsilon t \big) \\ &=&:J_1(x, t)+J_2(x, t). \end{eqnarray}$

$x_i\ge\gamma_i t(i=1, 2)$, 记$x_{i}^{\prime}=x_i+\mu _i\lambda t-\mu_i\lfloor \lambda t+\varepsilon t\rfloor\ge (1-{\varepsilon\mu_i}/{\gamma _i})x_i$. 因此, 由引理3.2和假设2.2可知

$\begin{eqnarray} J_1(x, t)&\le &{\rm P}\big(\vec{S}_{\lfloor \lambda t+\epsilon t \rfloor} -\vec{\mu }\lambda t>\vec{x}\big) \\ &=&{\rm P}\big( \vec{S}_{\lfloor \lambda t+\epsilon t \rfloor} -\vec{\mu}\lfloor \lambda t+\varepsilon t\rfloor >\vec{x}+\vec{\mu }\lambda t-\vec{\mu }\lfloor \lambda t+\varepsilon t \rfloor \big) \\ &\sim &\big(\lfloor\lambda t+\varepsilon t\rfloor\big)^2\overline{F}_1( x_{1}^{\prime})\overline{F}_2(x_{2}^{\prime}) \\ &\lesssim& \big(\lambda t+\varepsilon t\big)^2\overline{F}_1\big( ( 1-\varepsilon/\gamma)x_1 \big) \overline{F}_2\big( (1-\varepsilon/\gamma) x_2 \big). \end{eqnarray}$

$\begin{eqnarray} \lim\limits_{\varepsilon \downarrow 0} \limsup\limits_{t\rightarrow \infty}\sup\limits_{\vec{x}\ge \vec{\gamma }t}\frac{J_1(x, t)}{(\lambda t)^2\overline{F}_1(x_1)\overline{F}_2(x_2)}\le 1. \end{eqnarray}$

$\begin{eqnarray} J_2(x, t)&=&\sum\limits_{n>\lambda t+\varepsilon t}{{\rm P}\big(\vec{S}(t)-\vec{\mu }\lambda t>\vec{x}, N(t)=n\big)} \\ &\le&\sum\limits_{n>\lambda t+\varepsilon t}{{\rm P}\Big(\sum\limits_{k=1}^n\vec{X}_k>\vec{x}, \tau_n\le t\Big)} \\ &\le &C\overline{F}_1(x_1) \overline{F}_2(x_2) \sum\limits_{n>\lambda t+\varepsilon t}{ n^{2p+2}{\rm P}( \tau_{n-2d-1}\le t)}, \end{eqnarray}$

$\begin{eqnarray} \limsup\limits_{t\rightarrow \infty}\sup\limits_{\vec{x}\ge \vec{\gamma }t}\frac{J_2(x, t)}{(\lambda t)^2\overline{F}_1(x_1)\overline{F}_2(x_2)}=0, \end{eqnarray}$

$\begin{eqnarray} &&{\sum\limits_{\lambda t-\varepsilon t\le n\le \lambda t+\varepsilon t}}I_1(x, t) \\ &\ge& {\rm P} \bigg({\sum\limits_{i=3}^{\lambda t-\varepsilon t}}{X_{1i}-\mu _1\lambda t>( 1-2\nu ) x_1}, {\sum\limits_{i=3}^{\lambda t-\varepsilon t}}{X_{2i}-\mu _2\lambda t> ( 1-2\nu ) x_2}, \\ &&\Big|\frac{N(t)^{\ast\ast}-\lambda t}{t}\Big|\le \varepsilon \bigg){(\lfloor\lambda t-\varepsilon t\rfloor) (\lfloor\lambda t-\varepsilon t\rfloor-1)} \overline{F}_1(\nu x_1) \overline{F}_2(\nu x_2) \\ &\ge& \bigg({\rm P}\Big({\sum\limits_{i=3}^{\lambda t-\varepsilon t}}X_{1i}-\mu_1\lambda t>(1-2\nu)x_1, {\sum\limits_{i=3}^{\lambda t-\varepsilon t}}X_{2i}-\mu _2\lambda t>(1-2\nu)x_2\Big) \\ &&-{\rm P}\Big(\Big| \frac{N(t)^{\ast\ast}-\lambda t}{t}\Big|>\varepsilon \Big)\bigg)\cdot{\big(\lfloor\lambda t-\varepsilon t\rfloor\big)\big(\lfloor\lambda t-\varepsilon t\rfloor-1\big)} \overline{F}_1(\nu x_1)\overline{F}_2(\nu x_2). \end{eqnarray}$

$\begin{eqnarray} \lim\limits_{\varepsilon\searrow 0}\lim\limits_{\nu\searrow 1}\liminf\limits_{t\rightarrow \infty}\inf\limits_{\vec{x}\ge \vec{\gamma }t}\frac{\sum\limits_{\lambda t-\varepsilon t\le n\le \lambda t+\varepsilon t}I_1( x, t)}{(\lambda t)^2\overline{F}_1(x_1)\overline{F}_2(x_2)}\ge 1. \end{eqnarray}$

$I_2(x, t)$, 交换求和顺序后可以得到

$\begin{eqnarray} &&\sum\limits_{\lambda t-\varepsilon t \le n\le \lambda t+\varepsilon t}I_2(x, t) \\ &\le &\sum\limits_{j_1=1}^{\lambda t+\varepsilon t}\sum\limits_{1\le j_2\ne j_1\le \lambda t+\varepsilon t}{\sum\limits_{\lambda t-\varepsilon t\le n\le \lambda t+\varepsilon t}}{\rm P}(N(t)=n|X_{1j_1}>\nu x_1, X_{2j_1}>\nu x_2, X_{2j_2}>\nu x_2) \\ &&\cdot\overline{F}_{12}(\nu x_1, \nu x_2)\overline{F}_2(\nu x_2) \\ &&+\sum\limits_{i=1}^{\lambda t+\varepsilon t}\sum\limits_{j_1=1}^{\lambda t+\varepsilon t}\sum\limits_{j_2=1}^{\lambda t+\varepsilon t}{\sum\limits_{\lambda t-\varepsilon t\le n\le \lambda t+\varepsilon t}}{\rm P}( N(t) =n|X_{1i}>\nu x_1, X_{2j_1}>\nu x_2, X_{2j_2}>\nu x_2) \\ &&\cdot{\rm P}( X_{1i}>\nu x_1, X_{2j_2}>\nu x_2)\cdot\overline{F}_2(\nu x_2) \\ &\le&{(\lfloor\lambda t-\varepsilon t\rfloor)^2}\overline{F}_{12}(\nu x_1, \nu x_2) \overline{F}_2(\nu x_2)+{(\lfloor\lambda t-\varepsilon t\rfloor)}\overline{F}_2(\nu x_2) \\ &&\cdot\Big((\lfloor\lambda t-\varepsilon t\rfloor)\overline{F}_{12}(\nu x_1, \nu x_2)+{(\lfloor\lambda t-\varepsilon t\rfloor)(\lfloor\lambda t-\varepsilon t\rfloor -1)}\overline{F}_1(\nu x_1)\overline{F}_2(\nu x_2 )\Big) \\ &=&o\big((\lambda t)^2\overline{F}_1(x_1)\overline{F}_2(x_2)\big). \end{eqnarray}$

$\begin{eqnarray} \lim\limits_{\varepsilon\searrow 0}\lim\limits_{\nu\searrow 1}\limsup\limits_{t\rightarrow \infty}\sup\limits_{\vec{x}\ge \vec{\gamma }t}\frac{\sum\limits_{\lambda t-\varepsilon t\le n\le \lambda t+\varepsilon t}I_2( x, t)}{(\lambda t)^2\overline{F}_1(x_1)\overline{F}_2(x_2)}=0. \end{eqnarray}$

$\begin{eqnarray} \lim\limits_{\varepsilon\searrow 0}\lim\limits_{\nu\searrow 1}\limsup\limits_{t\rightarrow \infty}\sup\limits_{\vec{x}\ge \vec{\gamma }t }\frac{\sum\limits_{\lambda t-\varepsilon t\le n\le \lambda t+\varepsilon t}I_3( x, t)}{(\lambda t)^2\overline{F}_1(x_1)\overline{F}_2(x_2)}=0. \end{eqnarray}$

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