考虑一般投资收益和时间相依索赔情形下二维带扰动风险模型的有限时间破产概率渐近估计

考虑一般投资收益和时间相依索赔情形下二维带扰动风险模型的有限时间破产概率渐近估计

程铭^,, 王定成^,^*

电子科技大学数学科学学院成都 611731

Asymptotic Finite-Time Ruin Probability for a Bidimensional Perturbed Risk Model with General Investment Returns and Time-Dependent Claim Sizes

Cheng Ming^,, Wang Dingcheng^,^*

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731

通讯作者: * 王定成，Email: wangdc@uestc.edu.cn

收稿日期: 2021-06-21 修回日期: 2022-07-5

基金资助:

国家自然科学基金(71271042)
云南师范大学博士科研启动项目(2020ZB014)
云南省基础研究青年项目(202201AU070051)

Received: 2021-06-21 Revised: 2022-07-5

Fund supported:

NSFC(71271042)
Yunnan Normal University(2020ZB014)
Yunnan Province Science and Technology Department(202201AU070051)

作者简介 About authors

程铭，Email:chming@std.uestc.edu.cn

摘要

考虑具有一般投资收益过程的二维带扰动保险风险模型, 假定保险公司盈余的投资收益过程由右连左极随机过程刻画, 且两种索赔额与索赔到达时间间隔服从 Sarmanov 相依结构. 当索赔额分布属于正则变化尾分布族时, 得到有限时间破产概率的渐近公式. 当描述投资收益过程的右连左极过程分别取 Lévy 过程, Vasicek 利率模型, Cox-Ingersoll-Ross(CIR) 利率模型, Heston 模型时, 得到相应投资收益情形下破产概率的渐近公式.

关键词： 风险模型; 投资收益; 时间相依; 破产概率

Abstract

The paper considers a bi-dimensional perturbed insurance risk model with general investment returns. Assume that the investment return is described by a càdlàg process, and two classes of claims and the inter-arrival times follow the Sarmanov dependence structure. When the claim-size distribution has a regularly varying tail, the paper derives the asymptotic formula of the finite-time ruin probability. When the càdlàg process describing investment returns is chosen as the Lévy process, Vasicek interest rate model, Cox-Ingersoll-Ross (CIR) interest rate model, or Heston model, the paper derives the asymptotic estimates for ruin probabilities under the corresponding investment returns.

Keywords： Risk model; Investment return; Time-dependence; Ruin probability

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本文引用格式

程铭, 王定成. 考虑一般投资收益和时间相依索赔情形下二维带扰动风险模型的有限时间破产概率渐近估计[J]. 数学物理学报, 2023, 43(5): 1529-1558

Cheng Ming, Wang Dingcheng. Asymptotic Finite-Time Ruin Probability for a Bidimensional Perturbed Risk Model with General Investment Returns and Time-Dependent Claim Sizes[J]. Acta Mathematica Scientia, 2023, 43(5): 1529-1558

1 引言

近年来, 一维风险模型破产概率的研究成果丰硕^{[1,8,10,11,19,21,23]}. 对于保险公司来说, 只经营一种业务是不大可能的, 因此多维保险风险模型的研究被提上日程, 而二维保险风险模型作为一维保险风险模型的自然推广及高维保险风险模型的特例具有重要意义, 见文献 [3,4,5,6,7,9,12,16,17,18,20,25,27]. 随着我国财富日益增多和金融活动日益频繁, 保险公司资产的投资收益对其风险具有越来越重要的影响, 故在度量保险公司的破产概率时须考虑到资产的投资收益这一重要因素. 很多学者假设保险公司资产的投资收益是常数, 但即使保险公司在债券市场投资, 其收益率也是变化的, 这一理想化假设只是为了方便在数学上处理. 在金融活动日益频繁的今天, 保险公司通常会将投资组合的一部分资产购买可能获得较高回报的风险资产, 因此, 保险公司资产的投资收益具有不确定性与风险性. 也有学者假设保险公司资产的对数投资收益过程由 Lévy 过程刻画, 但对保险公司来说, 其投资收益过程并非全是由平稳独立过程刻画, 例如, CIR 利率模型刻画的投资收益过程是 Markov 过程, 基于分形布朗运动的随机波动性模型刻画的投资收益过程甚至不能由 Markov 过程刻画. 为保证保险公司破产概率的估计不会过低或过高, 如何选择随机过程描述保险公司资产的投资收益具有挑战性, 为解决这一问题, 假设保险公司的对数投资收益过程由一般的右连左极随机过程刻画, 本文研究二维带扰动保险风险模型破产概率的渐近估计.

设右连左极随机过程 $\{\xi(t), t\geq0\}$ 描述保险公司的对数投资收益过程且 $\xi(0)=0$ . 设 $\{B_1(t),$ $t\geq0\}$ 和 $\{B_2(t),t\geq0\}$ 是标准布朗运动且满足

$\begin{align*} B_2(t)=\rho B_1(t)+\sqrt{1-\rho^2}B_3(t), \end{align*}$

其中, $\rho\in[-1,1]$ , $B_3(t)$ 是标准布朗运动且与 $B_1(t)$ 独立. 于是, 截止时间 $t\geq0$ 保险公司的盈余过程 $(U_{1}(t),U_{2}(t))$ 为

$\begin{matrix} \begin{pmatrix} U_{1}(t) \\ U_{2}(t) \end{pmatrix} = \mathrm{e}^{\xi(t)}\left[ \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} c_1 \int_0^t \mathrm{e}^{-\xi(s)} \mathrm{d}s \\ c_2 \int_0^t \mathrm{e}^{-\xi(s)} \mathrm{d}s \end{pmatrix} + \begin{pmatrix} \sigma_1 \int_0^t \mathrm{e}^{-\xi(s)} \mathrm{d}B_1(s) \\ \sigma_2 \int_0^t \mathrm{e}^{-\xi(s)} \mathrm{d}B_2(s) \end{pmatrix} - \begin{pmatrix} \sum\limits_{i=1}^{N(t)} X_{i}\mathrm{e}^{-\xi(\tau_i)} ] \sum\limits_{j=1}^{N(t)} Y_{j}\mathrm{e}^{-\xi(\tau_j)} \end{pmatrix} \right], \end{matrix}$

(1.1)

其中, $(x,y)$ 是初始准备金向量; $(c_1, c_2)$ 是常数保费率向量且 $c_1, c_2>0$ ; $\sigma_i\geq0$ , $i=1,2$ , 是扰动系数; $\{(X_i, Y_i), i\geq1\}$ 是索赔额向量序列, 其共同到达时间间隔 $\theta_1$ , $\theta_2$ , $\cdots$ 是独立同分布且参数为 $\lambda>0$ 的指数分布随机变量; 到达时间 $\tau_n=\sum\limits_{i=1}^n \theta_i$ , $n\geq1$ 构成更新函数为 $\lambda_t=\lambda t$ 的齐次泊松过程 $\{N(t):N(t)=\sum\limits_{i=1}^\infty\mathbb{I}_{[\tau_i\leq t]}, t\geq0\}$ , 其中 $\mathbb{I}_{[A]}$ 表示集合 $A$ 的示性函数. 针对模型 (1.1), 本文定义有限时间破产概率为

$\begin{align*} \Psi \left(x,y;t\right) =P\left(T_{\text{max}}\leq t \mid U_{1}(0)=x, U_{2}(0)=y \right), \end{align*}$

其中 $T_{\text{max}}=\inf{\left\{t>0: \max{(U_{1}(t), U_{2}(t))}<0 \right\}}$ 是破产时间.

本文的行文组织架构如下. 第 2 节介绍一些预备知识并呈现本文主要结果; 当描述投资收益的随机过程分别取 Lévy 过程, Vasicek 利率模型, CIR 利率模型, Heston 模型时, 在第 3 节得到相应投资收益情形下破产概率的渐近公式; 第 4 节证明了本文主要结果.

2 准备知识及主要结果

本文约定 $C$ 总是表示一个正的常数, 且其在每处的大小可能不同. 对于实数 $a$ 和 $b$ , 记 $a \vee b=\max\{a,b\}$ , $a \wedge b=\min\{a,b\}$ . 对于满足关系

$\begin{align*} l_1 =\liminf_{(x,y)\rightarrow(\infty,\infty)} \frac{f(x,y)}{g(x,y)} \leq \limsup\limits_{(x,y)\rightarrow(\infty,\infty)} \frac{f(x,y)}{g(x,y)} = l_2, \end{align*}$

的两个正函数 $f(\cdot,\cdot)$ 和 $g(\cdot,\cdot)$ , 当 $l_2\leq1$ 时, 称 $f(x,y)\lesssim g(x,y)$ 成立; 当 $l_1\geq1$ 时, 称 $f(x,y)\gtrsim g(x,y)$ 成立; 当 $l_1=l_2=1$ 时, 称 $f(x,y)\sim g(x,y)$ 成立; 当 $0<l_1 \leq l_2< \infty$ 时, 称 $f(x,y)\asymp g(x,y)$ 成立.

定义2.1 称定义在 $[0,\infty)$ 上的分布函数 $F$ 是指数为 $-\alpha$ 的正则变化尾分布, 记作 $F\in\mathcal{R}_{-\alpha}$ , $\alpha\geq0$ , 若其生存函数 $\bar{F}(x)$ 满足

$\begin{matrix} \lim\limits_{x\rightarrow\infty} \frac{ \bar{F}(xy) }{ \bar{F}(x) } =y^{-\alpha} \end{matrix}$

(2.1)

对任意的 $y>0$ 成立.

本文假设索赔额及其共同到达时间间隔服从三元 Sarmanov 相依结构, 即

假设2.1 设 $(X, Y, \theta)$ 是独立同分布随机序列 $\left\{(X_i, Y_i, \theta_i), i\geq1\right\}$ 的独立复制, 其边际分布函数分别为定义在 $[0,\infty)$ 上的 $F$ , $G$ 和 $H$ 且满足

$\begin{matrix} &P(X \in \mathrm{d}x, Y \in \mathrm{d}y, \theta\in \mathrm{d}z)\nonumber\\ =\ &(1+\eta_{12}\varphi_{1}(x)\varphi_{2}(y)+\eta_{13}\varphi_{1}(x)\varphi_{3}(z) +\eta_{23}\varphi_{2}(y)\varphi_{3}(z))\mathrm{d}F(x)\mathrm{d}G(y)\mathrm{d}H(z), \end{matrix}$

(2.2)

其中, 参数 $\eta_{12}$ , $\eta_{13}$ 和 $\eta_{23}$ 皆为实数; 核函数 $\varphi_{1}(x)$ , $\varphi_{2}(y)$ 和 $\varphi_{3}(z)$ 满足

$\begin{equation} E \left( \varphi_{1}(X)\right) =E \left( \varphi_{2}(Y)\right) =E \left( \varphi_{3}(Z)\right) =0, \end{equation}$

(2.3)

$\begin{equation} \lim\limits_{x\rightarrow\infty} \varphi_1(x)=d_1<\infty,\quad \lim\limits_{y\rightarrow\infty} \varphi_2(y)=d_2<\infty,\quad \widetilde{C}_d:=1+\eta_{12}d_1d_2>0, \end{equation}$

(2.4)

$\begin{equation} 1+\eta_{12}\varphi_{1}(x)\varphi_{2}(y)+\eta_{13}\varphi_{1}(x)\varphi_{3}(z)+\eta_{23}\varphi_{2}(y)\varphi_{3}(z)\geq0, \quad \text{对} \ x,y,z\geq0. \end{equation}$

(2.5)

微分等式 (2.2) 两边关于 $z$ 在区间 $[0,\infty)$ 取积分, 根据 (2.3) 式得

$\begin{equation} P(X \in \mathrm{d}x, Y \in \mathrm{d}y) =\left(1+\eta_{12}\varphi_{1}(x)\varphi_{2}(y)\right) \mathrm{d}F(x)\mathrm{d}G(y), \end{equation}$

(2.6)

即随机向量 $(X,Y)$ 服从二元 Sarmanov 分布. 类似地,

$\begin{matrix} P(X \in \mathrm{d}x, \theta\in \mathrm{d}z) =(1+\eta_{13}\varphi_{1}(x)\varphi_{3}(z)) \mathrm{d}F(x)\mathrm{d}H(z), \end{matrix}$

(2.7)

$\begin{matrix} P(Y \in \mathrm{d}y, \theta\in \mathrm{d}z) =(1+\eta_{23}\varphi_{2}(y)\varphi_{3}(z)) \mathrm{d}G(y)\mathrm{d}H(z). \end{matrix}$

(2.8)

根据文献^[26], 存在常数 $b_1$ , $b_2$ 和 $b_3$ 使得

$\begin{matrix} \vert \varphi_1(x)\vert\leq b_1,\quad \vert \varphi_2(y)\vert\leq b_2,\quad \vert \varphi_3(z)\vert\leq b_3, \qquad \text{对} \ x,y,z\geq 0. \end{matrix}$

(2.9)

接下来首先引入一些记号. 由 (2.4) 式, 对 $s>0$ , 定义 $\varphi_{31}(s)$ , $\varphi_{32}(s)$ 和 $\varphi_{33}(s)$ ,

$\begin{matrix} &\varphi_{31}(s) := 1+\eta_{13}d_1\varphi_{3}(s)\geq0, \quad \varphi_{32}(s) := 1+\eta_{23}d_2\varphi_{3}(s)\geq0, \end{matrix}$

(2.10)

$\begin{matrix} \varphi_{33}(s) := 1+\varphi_3(s)(\eta_{13}d_1+\eta_{23}d_2)/\widetilde{C}_d\geq0. \end{matrix}$

(2.11)

根据 $\varphi_3(\cdot)$ 的有界性, 存在正的常数 $b_{31}$ , $b_{32}$ 和 $b_{33}$ 使得 $\varphi_{31}(\cdot)\leq b_{31}$ , $\varphi_{32}(\cdot)\leq b_{32}$ 和 $\varphi_{33}(\cdot)\leq b_{33}$ 成立. 令 $\widehat{\theta}_1$ , $\widehat{\theta}_2$ 和 $\widehat{\theta}_3$ 是独立的随机变量, 且分布函数分别为 $\widehat{H}_1$ , $\widehat{H}_2$ 和 $\widehat{H}_3$ , 即

$\begin{matrix} \mathrm{d}\widehat{H}_1(s)=\varphi_{31}(s)\mathrm{d}H(s),\quad \mathrm{d}\widehat{H}_2(s)=\varphi_{32}(s)\mathrm{d}H(s),\quad \mathrm{d}\widehat{H}_3(s)=\varphi_{33}(s)\mathrm{d}H(s). \end{matrix}$

(2.12)

设随机向量 $(X^{\,*}, Y^{\,*}, \theta^*)$ 是随机向量 $(X, Y, \theta)$ 的独立版本, 也就是说, 随机向量 $(X^{\,*}, Y^{\,*}, \theta^*)$ 与随机向量 $(X, Y, \theta)$ 具有相同的分布且随机向量 $(X^{\,*}, Y^{\,*}, \theta^*)$ 的元素之间相互独立. 同时, 设 $\{(X_k^{\,*},$ $Y_k^{\,*}, \theta_k^*), k\geq1\}$ 是独立复制版本为 $(X^{\,*}, Y^{\,*}, \theta^*)$ 的独立同分布的随机向量序列. 于是, 本节构造下列更新过程,

$\begin{matrix} \widehat{N}_t^{(m)} =\sum\limits_{k=1}^{\infty} \mathbb{I}_{(\widehat{\tau}_{k}^{(m)}\leq t)},\ \ \widehat{\tau}_{1}^{(m)}=\widehat{\theta}_m,\ \widehat{\tau}_{k}^{(m)}=\widehat{\theta}_m+\sum\limits_{i=2}^{k}\theta_i^*,\ k\geq 2,\ m=1,2,3;\end{matrix}$

(2.13)

$\begin{matrix}\widehat{N}_t^{(4)}=\sum\limits_{k=1}^{\infty}\mathbb{I}_{(\widehat{\tau}_{k}^{(4)}\leq t)},\ \ \widehat{\tau}_{1}^{(4)}=\widehat{\theta}_1,\ \widehat{\tau}_{2}^{(4)}=\widehat{\theta}_1+\widehat{\theta}_2,\ \widehat{\tau}_{k}^{(4)}=\widehat{\theta}_1+\widehat{\theta}_2+\sum\limits_{i=3}^{k}\theta_i^*, \ k\geq 3;\end{matrix}$

(2.14)

$\begin{matrix}\widehat{N}_t^{(5)}=\sum\limits_{k=1}^{\infty}\mathbb{I}_{(\widehat{\tau}_{k}^{(5)}\leq t)},\ \ \widehat{\tau}_{1}^{(5)}=\widehat{\theta}_2,\ \widehat{\tau}_{2}^{(5)}=\widehat{\theta}_2+\widehat{\theta}_1,\ \widehat{\tau}_{k}^{(5)}=\widehat{\theta}_2+\widehat{\theta}_1+\sum\limits_{i=3}^{k}\theta_i^*, \ k\geq 3. \end{matrix}$

(2.15)

且更新函数为

$\begin{matrix} \widehat{\lambda}_{t}^{(m)} =E\widehat{N}_{t}^{(m)} =\sum\limits_{k=1}^{\infty}P(\widehat{\tau}_k^{(m)}\leq t),\qquad m=1,2,3,4,5. \end{matrix}$

(2.16)

本文假设随机量 $\left\{(X_i, Y_i, \theta_i), i\geq1\right\}$ , $\left\{(X_i^{\,*}, Y_i^{\,*}, \theta_i^*), i\geq1\right\}$ , $\left\{\xi(t), t\geq0\right\}$ , $\left\{(B_1(t),B_2(t)), t\geq0\right\}$ 和 $(\widehat{\theta}_1, \widehat{\theta}_2, \widehat{\theta}_3)$ 独立.

定理2.1 考虑二维保险风险模型 (1.1). 设 $F\in\mathcal{R}_{-\alpha}$ , $G \in \mathcal{R}_{-\beta}$ , 其中 $\alpha>0$ , $\beta>0$ . 在假设 2.1 成立的条件下, 若存在常数 $\kappa > \max\{ \alpha+\beta, 2 \}$ , 使得 $\int_{0}^{T}E\mathrm{e}^{-\kappa\xi(s)}\mathrm{d}s<\infty$ 对 $0<T<\infty$ 成立, 则对 $0< t\leq T$ , $\Psi(x,y;t) \sim \bar{F}(x)\bar{G}(y)P(t),$ 其中

$\begin{align*} P(t) =\ & \widetilde{C}_d \int_{0}^{t}E\mathrm{e}^{-(\alpha+\beta)\xi(u)}\mathrm{d}\widehat{\lambda}_{u}^{(3)} \\ &+\iint_{u+v\leq t} E\mathrm{e}^{-\beta\xi(u)-\alpha\xi(u+v)} [\mathrm{d}\widehat{\lambda}_{u}^{(5)}\mathrm{d}\lambda_v+\mathrm{d}\widehat{H}_2(u)(\mathrm{d}\widehat{\lambda}_{v}^{(1)}-\mathrm{d}\lambda_v)] \\ & +\iint_{u+v\leq t} E\mathrm{e}^{-\alpha\xi(u)-\beta\xi(u+v)} [\mathrm{d}\widehat{\lambda}_{u}^{(4)}\mathrm{d}\lambda_v+\mathrm{d}\widehat{H}_1(u)(\mathrm{d}\widehat{\lambda}_{v}^{(2)}-\mathrm{d}\lambda_v)]. \end{align*}$

当对数投资收益过程 $\{\xi(t),t\geq0\}$ 是 Lévy 过程时, 显然有推论 2.1.

推论2.1 考虑二维保险风险模型 (1.1). 设 $\{\xi(t),t\geq0\}$ 是 Lévy 过程, $\phi(\cdot)$ 是 $\{\xi(t),t\geq0\}$ 的 Laplace 指数. 在定理 (2.1) 的条件下, 则对 $0< t\leq T$ ,

$\Psi(x,y;t) \sim \bar{F}(x)\bar{G}(y)P'(t),$

其中

$\begin{align*} P'(t)&= \widetilde{C}_d \int_{0}^{t}\mathrm{e}^{u\phi(\alpha+\beta)}\mathrm{d}\widehat{\lambda}_{u}^{(3)} +\iint_{u+v\leq t} \mathrm{e}^{u\phi(\alpha+\beta)+v\phi(\alpha)} [\mathrm{d}\widehat{\lambda}_{u}^{(5)}\mathrm{d}\lambda_v+\mathrm{d}\widehat{H}_2(u)(\mathrm{d}\widehat{\lambda}_{v}^{(1)}-\mathrm{d}\lambda_v)] \nonumber\\ &\quad\ +\iint_{u+v\leq t} \mathrm{e}^{u\phi(\alpha+\beta)+v\phi(\beta)} [\mathrm{d}\widehat{\lambda}_{u}^{(4)}\mathrm{d}\lambda_v+\mathrm{d}\widehat{H}_1(u)(\mathrm{d}\widehat{\lambda}_{v}^{(2)}-\mathrm{d}\lambda_v)]. \end{align*}$

设随机向量 $(X,Y)$ 服从 (2.6) 式中的二元 Sarmanov 分布, 且 $(X,Y\,)$ 和 $\theta$ 独立, 即在 (2.2) 式中有 $\eta_{13}=\eta_{23}=0$ , $\varphi_3(z)=0$ . 于是由 (2.12) 式, $\widehat{H}_1=\widehat{H}_2=\widehat{H}_3=H$ 且 $\widehat{\lambda}_t^{(m)}=\lambda_t$ , $m=1,2,3,4,5$ , 对任意的 $t>0$ 成立. 故得到下面的推论.

推论2.2 考虑二维保险风险模型 (1.1). 设随机向量 $(X,Y)$ 服从 (2.6) 式中的二元 Sarmanov 分布且其边际分布为 $F,G\in\mathcal{R}_{-\alpha}$ , $\alpha>0$ . 设随机向量 $(X,Y\,)$ 与随机变量 $\theta$ 独立. 在假设 2.1 成立的条件下, 若存在常数 $\kappa > \max\{ 2\alpha, 2 \}$ 使得 $\int_{0}^{T}E\mathrm{e}^{-\kappa\xi(s)}\mathrm{d}s<\infty$ 对 $0<T<\infty$ 成立, 则对 $0<t\leq T$ ,

$\begin{align*} \Psi(x,y;t) \sim \bar{F}(x)\bar{G}(y) &\left\{ \lambda\widetilde{C}_d \int_{0}^{t}E\mathrm{e}^{-2\alpha\xi(u)}\mathrm{d}u +2\lambda^2\iint_{u+v\leq t} E\mathrm{e}^{-\alpha\xi(u)-\alpha\xi(u+v)} \mathrm{d}u\mathrm{d}v \right\}. \end{align*}$

3 两个应用

当对数投资收益过程 $\{\xi(t),t\geq0\}$ 是 Vasicek 模型、CIR 模型或 Heston 模型刻画的右连左极随机过程时, 本节将计算二维保险风险模型 (1.1) 的有限时间破产概率的渐近公式. 根据定理 (2.1), 计算有限时间破产概率渐近公式的关键是计算 $E\mathrm{e}^{-b\xi(u)-a\xi(u+v)}$ , 其中 $a$ , $b$ 为实数. 于是, 本节重点讨论 $E\mathrm{e}^{-b\xi(u)-a\xi(u+v)}$ .

3.1 应用于 Vasicek 模型和 CIR 模型

设对数投资收益过程 $\{\xi(t),\ t\geq 0\}$ 满足

$\begin{matrix} \xi(t)=\int_0^t r_s \mathrm{d}s, \quad \xi(0)=0, \end{matrix}$

(3.1)

$\begin{matrix} \mathrm{d}r_t=m(l-r_t)\mathrm{d}t+\delta r_t^{\pi}\mathrm{d}W_t, \end{matrix}$

(3.2)

其中, $\{r_t,\ t\geq 0\}$ 是短期随机利率过程; $m$ , $l$ 和 $\delta$ 皆为正的常数; $\pi=0$ 或 $1/2$ ; $\{W_t,\ t\geq 0\}$ 是标准布朗运动. 在金融数学中, 当 $\pi=0$ 时模型 (3.2) 被称为 Vasicek 模型, 当 $\pi=1/2$ 时模型 (3.2) 被称为 CIR 模型. 与 CIR 模型相比, Vasicek 模型中不存在平方根项, 于是 Vasicek 模型会出现负利率.

引理3.1 设右连左极随机过程 $\{\xi(t),t\geq 0\}$ 满足 (3.1) 和 (3.2) 式, 则对实数 $a$ 和 $b$ ,

(1) 当 $\pi=0$ 即 Vasicek 模型时, 有

$\begin{matrix} E\mathrm{e}^{-b\xi(u)-a\xi(u+v)} &=\exp{\left\{ \widehat{A}_1(a,b)+\widehat{A}_2(a,b)v+\widehat{A}_3(a,b)\mathrm{e}^{-mv}\right\}}\nonumber\\ &\quad\cdot\exp{\left\{ \widehat{A}_4(a,b)u +\widehat{A}_5(a,b)\mathrm{e}^{-mu}+\widehat{A}_6(a,b)\mathrm{e}^{-2mu}\right\}}\nonumber\\ &\quad\cdot\exp{\left\{ \widehat{A}_7(a,b)\mathrm{e}^{-mu-mv}+\widehat{A}_8(a,b)\mathrm{e}^{-2mu-mv}+\widehat{A}_{9}(a,b)\mathrm{e}^{-2mu-2mv}\right\}}, \end{matrix}$

(3.3)

其中

$\widehat{A}_1(a,b)=\frac{a+b}{m}(l-r_0)-\frac{3a^2+3b^2+4ab}{4m^3}\delta^2, \ \widehat{A}_2(a,b)=-al+\frac{a^2\delta^2}{2m^2},\ \widehat{A}_3(a,b)=\frac{-ab\delta^2}{2m^3},$

$\widehat{A}_4(a,b)=(a+b)\left(\frac{a+b}{2m^2}\delta^2-l\right),\ \widehat{A}_5(a,b)=\frac{b}{m}\left(r_0-l+\frac{(a+b)\delta^2}{m^2}\right),\ \widehat{A}_6(a,b)=-\frac{b^2\delta^2}{4m^3},$

$\widehat{A}_7(a,b)=\frac{a}{m}\left(r_0-l+\frac{(a+b)\delta^2}{m^2}\right),\ \widehat{A}_8(a,b)=-\frac{ab\delta^2}{2m^3},\ \widehat{A}_9(a,b)=-\frac{a^2\delta^2}{4m^3}.$

(2) 当 $\pi=1/2$ 即 CIR 模型时, 若 $m^2+2\delta^2[(a+b)\wedge a]>0$ , 则

$\begin{matrix} E\mathrm{e}^{-b\xi(u)-a\xi(u+v)} \\ =&\left(\frac{k_a\mathrm{e}^{mv/2}}{k_a\cosh(k_av)\!+\!\sinh(k_av)m/2}\right)^{2aml/\delta^2} \left(\frac{(\widehat{B}_1(v)\!-\!1)\mathrm{e}^{(m-2k_{a+b})u/2}}{\widehat{B}_1(v)\!-\!\mathrm{e}^{-2uk_{a+b}}}\right)^{2ml/\delta^2}\nonumber\\ & \cdot \exp{\left\{r_0\left[\frac{m-2k_{a+b}}{\delta^2}-\frac{4k_{a+b}}{\delta^2[\widehat{B}_1(v)\mathrm{e}^{2uk_{a+b}}-1]}\right]\right\}}, \end{matrix}$

(3.4)

其中 $k_s=\sqrt{m^2+2s\delta^2}/2$ , $\widehat{B}_1(v) =1-\frac{4k_{a+b}}{(2k_{a+b}-m)-\widehat{B}_2(v)\delta^2}$ , $\widehat{B}_2(v)=\frac{2a}{m+2k_a\coth(k_av)}$ .

证由 (3.1) 式,

$\begin{align*} E\mathrm{e}^{-b\xi(u)-a\xi(u+v)} =E\left(\mathrm{e}^{-(a+b)\int_0^u r_s\mathrm{d}s -a\int_u^{u+v}r_s\mathrm{d}s}\right) =E\left\{\mathrm{e}^{-(a+b)\int_0^u r_s\mathrm{d}s} E\left(\mathrm{e}^{-a\int_u^{u+v}r_s\mathrm{d}s}|\mathcal{F}_u\right)\right\}. \end{align*}$

本证明的总体思路是首先计算 $E\big(\mathrm{e}^{-a\int_u^{u+v}r_sds}|\mathcal{F}_u\big)$ , 然后计算 $E\mathrm{e}^{-b\xi(u)-a\xi(u+v)}$ . $r(t)$ 是由随机微分方程 (3.2) 确定, 故 $r(t)$ 是马尔科夫过程, 且对 $\widetilde{T}\geq t$ 记 $B(t,\widetilde{T}\,):=E\big(\mathrm{e}^{-a\int_t^{\widetilde{T}}r_sds}|\mathcal{F}_t\big)=f\,(t,r_t)$ , 其中 $f\,(t,r)$ 是关于虚拟变量 $t$ 和 $r$ 的函数. 为得到 $E\big(\mathrm{e}^{-a\int_u^{u+v}r_sds}|\mathcal{F}_u\big)$ , 先求解 $f\,(t,r)$ 的显式表达式. 根据文献 [22], 需通过"建立鞅, 取微分, 令 $\mathrm{d}t$ 项等于 $0$ " 这三步找到函数 $f\,(t,r)$ 满足的偏微分方程, 通过求解偏微分方程得到 $f\,(t,r)$ 的显式表达式. 对 $0\leq s\leq t \leq \widetilde{T}$ , $E\left\{\mathrm{e}^{-a\xi(t)}\cdot B(t,\widetilde{T}\,)\Big|\mathcal{F}_s\right\} =E\left\{\mathrm{e}^{-a\int_0^tr_u\mathrm{d}u}\cdot E\big(\mathrm{e}^{-a\int_t^{\widetilde{T}}r_s\mathrm{d}s}|\mathcal{F}_t\big)|\mathcal{F}_s\right\} =E\left\{\mathrm{e}^{-a\int_0^{\widetilde{T}}r_u\mathrm{d}u}|\mathcal{F}_s\right\} =\mathrm{e}^{-a\xi(s)}\cdot B(s,\widetilde{T})$ . 故 $\mathrm{e}^{-a\xi(t)}\cdot B(t,\widetilde{T}\,)=\mathrm{e}^{-a\xi(t)}\cdot f\,(t,r_t)$ 是鞅. 鞅 $\mathrm{e}^{-a\xi(t)}\cdot f\,(t,r_t)$ 的微分为

$\begin{align*} \mathrm{d}\left(\mathrm{e}^{-a\xi(t)}\cdot f\,(t,r_t)\right) &=f\,(t,r_t)\mathrm{d}\mathrm{e}^{-a\xi(t)}+\mathrm{e}^{-a\xi(t)}\mathrm{d}f\,(t,r_t)\\ &=\mathrm{e}^{-a\xi(t)}\left\{-arf+f_t+m(l-r)f_r +f_{rr}\cdot\delta^2r^{2\pi}/2 \right\}\mathrm{d}t+\delta \mathrm{e}^{-a\xi(t)}f_r\,\mathrm{d}W_t. \end{align*}$

令 $\mathrm{d}t$ 项等于 $0$ 得到偏微分方程

$\begin{matrix} f_t(t,r)+m(l-r)f_r(t,r)+f_{rr}(t,r)\cdot\delta^2r^{2\pi}/2 =arf\,(t,r), \end{matrix}$

(3.5)

且终端条件为

$\begin{equation} f\,(\widetilde{T},r)=1,\quad \text{对任意的}\ r. \end{equation}$

(3.6)

(1) 当 $\pi=0$ , 即 Vasicek 模型. 偏微分方程 (3.5) 成为

$\begin{matrix} f_t(t,r)+m(l-r)f_r(t,r)+f_{rr}(t,r)\cdot\delta^2/2 =arf\,(t,r). \end{matrix}$

(3.7)

根据文献 [22], 该偏微分方程的解的形式为

$f\,(t,r)=\mathrm{e}^{-rC_1(t,\widetilde{T})-A_1(t,\widetilde{T})}$ , 其中 $C_1(t,\widetilde{T})$ 和 $A_1(t,\widetilde{T})$ 是关于 $t\in[\widetilde{T}]$ 的函数. 将 $f\,(t,r)=\mathrm{e}^{-rC_1(t,\widetilde{T})-A_1(t,\widetilde{T})}$ 代入偏微分方程 (3.7) 得到

$\begin{matrix} \left[\left( -C'_1(t,\widetilde{T})+ mC_1(t,\widetilde{T})- a\right)r -A'_1(t,\widetilde{T}) -mlC_1(t,\widetilde{T}) +\frac{\delta^2C_1^2(t,\widetilde{T})}{2}\right]f\,(t,r) =0. \end{matrix}$

(3.8)

对任意的 $r$ , (3.8) 式总是成立的, 故 $-C'_1(t,\widetilde{T})+mC_1(t,\widetilde{T})-a=0$ , 否则, 若改变 $r$ 的取值, (3.8) 式并不总是 $0$ . 于是得到常微分方程

$\begin{matrix} C'_1(t,\widetilde{T})=mC_1(t,\widetilde{T})-a. \end{matrix}$

(3.9)

从而,

$\begin{equation} A'_1(t,\widetilde{T})=-mlC_1(t,\widetilde{T})+\delta^2C_1^2(t,\widetilde{T})/2. \end{equation}$

(3.10)

由终端条件 (3.6), $C_1(\widetilde{T},\widetilde{T})=A_1(\widetilde{T},\widetilde{T})=0$ . 故根据 (3.9) 和 (3.10)} 式及终端条件可得

$\begin{equation} C_1(t,\widetilde{T})=\frac{a}{m}\left(1-\mathrm{e}^{-m(\widetilde{T}-t)}\right), \end{equation}$

(3.11)

$\begin{equation} A_1(t,\widetilde{T}) =\left(al-\frac{a^2\delta^2}{2m^2}\right)(\widetilde{T}-t) +\left(\frac{al}{m}-\frac{a^2\delta^2}{m^3}\right)\mathrm{e}^{-m(\widetilde{T}-t)} +\frac{a^2\delta^2}{4m^3}\mathrm{e}^{-2m(\widetilde{T}-t)} -\left(\frac{al}{m}-\frac{3a^2\delta^2}{4m^3}\right). \end{equation}$

(3.12)

于是, $B(t,\widetilde{T}\,)$ 的显式表达为 $B(t,\widetilde{T}\,)=f\,(t,r_t)=\mathrm{e}^{-r_tC_1(t,\widetilde{T})-A_1(t,\widetilde{T})}$ , $0\leq t\leq \widetilde{T}$ , 其中 $C_1(t,\widetilde{T})$ 和 $A_1(t,\widetilde{T})$ 分别由 (3.11) 和 (3.12) 式确定. 故

$\begin{matrix} E\mathrm{e}^{-b\xi(u)-a\xi(u+v)} &=E\left\{\mathrm{e}^{-(a+b)\xi(u)}\cdot B(u,u+v)\right\}\nonumber\\ &=\mathrm{e}^{-A_1(u,u+v)}E\left\{\mathrm{e}^{-C_1(u,u+v)r_u-(a+b)\xi(u)}\right\}. \end{matrix}$

(3.13)

要得到 $E\left\{\mathrm{e}^{-C_1(u,u+v)r_u-(a+b)\xi(u)}\right\}$ , 只需考虑 $E\mathrm{e}^{p_1r_t+p_2\xi(t)}$ , 其中 $p_1$ , $p_2$ 是实数. 令 $D(p_1,p_2,t)=E\mathrm{e}^{p_1r_t+p_2\xi(t)}$ , $r_t$ 是高斯随机变量, 由文献 [22,例 4.4.10] 知 $\xi(t)=\int_0^t r_s \mathrm{d}s=(mlt+r_0-r_t+\delta W_t)/m$ 也是高斯随机变量, 故对任意实数 $p_1$ 和 $p_2$ , $D(p_1,p_2,t)$ 是有限的. 对 $\mathrm{e}^{p_1r_t+p_2\int_0^t r_s \mathrm{d}s}$ 应用 Itô 公式并取期望得

$D(p_1,p_2,t) =\mathrm{e}^{p_1r_0}+\left(p_1ml+\frac{p_1^2\delta^2}{2}\right)\int_0^tD(p_1,p_2,s)\,\mathrm{d}s+(p_2-p_1m)\int_0^t\frac{\partial D(p_1,p_2,s)}{\partial p_1 } \,\mathrm{d}s,$

其中上式用到了矩母函数的性质 $\partial D(p_1,p_2,s) /\partial p_1 =E(r_s\mathrm{e}^{p_1r_s+p_2\int_0^s r_v \mathrm{d}v})$ . 于是得到 $D(p_1,p_2,t)$ 关于 $t$ 的积微分方程

$\begin{matrix} \frac{\partial D(p_1,p_2,t)}{\partial t } + (p_1m-p_2)\frac{\partial D(p_1,p_2,t)}{\partial p_1} =\left(p_1ml+\frac{p_1^2\delta^2}{2}\right)D(p_1,p_2,t), \end{matrix}$

(3.14)

且初始条件为 $D(p_1,p_2,0)=\mathrm{e}^{p_1r_0}$ . 要得到方程 (3.14) 的解, 首先要找到 (3.14) 式的特征线方程的解. 由特征线方程的定义,

$\begin{matrix} \frac{\mathrm{d}p_1}{\mathrm{d}s}=(p_1m-p_2),\quad \frac{\mathrm{d}t}{\mathrm{d}s}=1,\quad p_1(0)=c,\quad t(0)=0. \end{matrix}$

(3.15)

于是, (3.15) 式的解为

$\begin{matrix} p_1(s) =\frac{p_2}{m}+\left(c-\frac{p_2}{m}\right)\mathrm{e}^{ms},\quad t(s)=s. \end{matrix}$

(3.16)

取 $U(s)=D(p_1(s),p_2,t(s))$ , 于是 (3.14) 式可转化为下面的常微分方程,

$\begin{align*} \frac{\mathrm{d}U}{\mathrm{d}s} =\frac{\partial D(p_1,p_2,t)}{\partial t}\frac{\mathrm{d}t}{\mathrm{d}s} +\frac{\partial D(p_1,p_2,t)}{\partial p_1}\frac{\mathrm{d}p_1}{\mathrm{d}s} = \frac{\partial D(p_1,p_2,t)}{\partial t}+(p_1m-p_2)\frac{\partial D(p_1,p_2,t)}{\partial p_1}. \end{align*}$

结合 (3.14) 式, 则 $\frac{\mathrm{d}U}{\mathrm{d}s} =\left(p_1ml+\frac{p_1^2\delta^2}{2}\right)U(s)$ , 且初值条件为 $U(0)=D(p_1(0),p_2,t(0))=D(c,p_2,0)=\mathrm{e}^{cr_0}$ . 于是,

$\begin{matrix} U(s)=\exp{\left\{cr_0+ml\int_0^sp_1(v)\mathrm{d}v+\frac{\delta^2}{2}\int_0^sp_1^2(v)\mathrm{d}v\right\}}. \end{matrix}$

(3.17)

由 (3.16) 式, 根据 $p_1$ 和 $t$ 来表示 $c$ 和 $s$ , 随后代入 (3.17) 式得

$\begin{matrix} D(p_1,p_2,t) =\exp{\left\{\bar{A}_1(p_1,p_2)+\bar{A}_2(p_1,p_2)t+\bar{A}_3(p_1,p_2)\mathrm{e}^{-mt}+\bar{A}_4(p_1,p_2)\mathrm{e}^{-2mt}\right\}}, \end{matrix}$

(3.18)

其中 $\bar{A}_1(p_1,p_2) =\frac{p_2r_0}{m} +\left(l+\frac{p_2\delta^2}{m^2}\right) \left(p_1-\frac{p_2}{m}\right) +\left(p_1-\frac{p_2}{m}\right)^2\frac{\delta^2}{4m}$ , $\bar{A}_2(p_1,p_2) =p_2l+\frac{p_2^2\delta^2}{2m^2}$ , $\bar{A}_3(p_1,p_2)$ $=\left(r_0-l-\frac{p_2\delta^2}{m^2}\right) \left(p_1-\frac{p_2}{m}\right)$ , $\bar{A}_4(p_1,p_2) =-\frac{\delta^2(p_1-p_2/m)^2}{4m}$ . 事实上, 根据 (3.18) 式,

$\begin{matrix} E\left\{\mathrm{e}^{-C_1(u,u+v)r_u-(a+b)\xi(u)}\right\} &=E\left\{\mathrm{e}^{\frac{-a(1-\mathrm{e}^{-mv})}{m}r_u-(a+b)\xi(u)}\right\}\nonumber\\ &=D\left(-a\left(1-\mathrm{e}^{-mv}\right)/m,-(a+b),u\right). \end{matrix}$

(3.19)

故将 (3.19) 式代入 (3.13) 式即可得到 (3.3) 式.

(2) 当 $\pi=1/2$ , 即 CIR 模型. 偏微分方程 (3.5) 成为

$\begin{matrix} f_t(t,r)+m(l-r)f_r(t,r)+\delta^2rf_{rr}(t,r)/2=arf\,(t,r). \end{matrix}$

(3.20)

该偏微分方程的解具有形式 $f\,(t,r)=\mathrm{e}^{-rC_2(t,\widetilde{T})-A_2(t,\widetilde{T})}$ , 将其代入偏微分方程 (3.20) 得

$\begin{align*} f\,(t,r)\left[\left(-C'_2(t,\widetilde{T})+mC_2(t,\widetilde{T}) +\frac{\delta^2}{2}C_2^2(t,\widetilde{T})-a\right)r -A'_2(t,\widetilde{T})-mlC_2(t,\widetilde{T})\right]=0. \end{align*}$

类似地, 得到 $-C'_2(t,\widetilde{T})+mC_2(t,\widetilde{T}) +\frac{\delta^2}{2}C_2^2(t,\widetilde{T})-a=0$ 和 $-A'_2(t,\widetilde{T})-mlC_2(t,\widetilde{T})=0$ , 即 $C'_2(t,\widetilde{T})=mC_2(t,\widetilde{T})+\delta^2C_2^2(t,\widetilde{T})/2-a$ 和 $A'_2(t,\widetilde{T})=-mlC_2(t,\widetilde{T})$ . 而且, 该方程满足终端条件 $C_2(\widetilde{T},\widetilde{T})=A_2(\widetilde{T},\widetilde{T})=0$ . 首先求解 $C_2(t,\widetilde{T})$ . 令 $Z(t,\widetilde{T})=C_2(t,\widetilde{T})+(m+2k_a)/\delta^2$ , 其中 $k_a=\sqrt{m^2+2a\delta^2}/2$ . 于是, $Z'(t,\widetilde{T}) =\frac{\delta^2}{2}Z^2(t,\widetilde{T})-2k_aZ(t,\widetilde{T})$ 是终端条件为 $Z(\widetilde{T},\widetilde{T})=(m+2k_a)/\delta^2$ 的伯努利微分方程, 故 $Z(t,\widetilde{T}) =\left(\frac{\delta^2}{4k_a} +\delta^2\left[\frac{1}{m+2k_a}-\frac{1}{4k_a}\right] \mathrm{e}^{-2k_a(\widetilde{T}-t)}\right)^{-1}$ . 于是

$\begin{matrix} C_2(t,\widetilde{T}) =\frac{a\sinh(k_a(\widetilde{T}-t))}{ k_a\cosh(k_a(\widetilde{T}-t)) +\frac{m}{2}\sinh(k_a(\widetilde{T}-t))}. \end{matrix}$

(3.21)

然后求解 $A_2(t,\widetilde{T})$ . 对微分方程 $A'_2(t,\widetilde{T})=-mlC_2(t,\widetilde{T})$ 两端积分并利用终端条件 $A_2(\widetilde{T},\widetilde{T})=0$ 得

$\begin{gather} A_2(t,\widetilde{T}) =-\frac{2mla}{\delta^2}\ln\left( \frac{k_a\mathrm{e}^{m(\widetilde{T}-t)/2}}{ k_a\cosh(k_a(\widetilde{T}-t)) +\frac{m}{2}\sinh(k_a(\widetilde{T}-t)) }\right). \end{gather}$

(3.22)

于是, CIR 模型中 $f\,(t,r_t)$ 的显式解为 $B(t,\widetilde{T}\,) =f\,(t,r_t) =\mathrm{e}^{-r_tC_2(t,\widetilde{T})-A_2(t,\widetilde{T})}$ , $0\leq t\leq \widetilde{T}$ , 其中 $C_2(t,\widetilde{T})$ 和 $A_2(t,\widetilde{T})$ 由 (3.21) 和 (3.22) 式确定. 故

$\begin{matrix} E\left(\mathrm{e}^{-b\xi(u)-a\xi(u+v)}\right) &=E\left\{\mathrm{e}^{-(a+b)\xi(u)}\cdot B(u,u+v)\right\}\nonumber\\ &=\mathrm{e}^{-A_2(u,u+v)}E\left\{\mathrm{e}^{-C_2(u,u+v)r_u-(a+b)\xi(u)}\right\}. \end{matrix}$

(3.23)

为得到 $E\{\mathrm{e}^{-C_2(u,u+v)r_u-(a+b)\xi(u)}\}$ , 设 $D'(p_1,p_2,t)=E\mathrm{e}^{p_1r_t+p_2\xi(t)}$ , 其中 $p_1$ 和 $p_2$ 是实数. 由文献 [13], $r_t$ 和 $\xi(t)=\int_0^tr_s\mathrm{d}s$ 皆有矩母函数, 类似于上面与 $D(p_1,p_2,t)$ 有关的讨论知, 当 $m^2>2\delta^2p_2$ 时,

$\begin{matrix} D'(p_1,p_2,t) =\exp{\left\{\hat{c}(p_1,p_2)r_0+\frac{[m-\Omega(p_2)]mlt}{\delta^2}- \frac{2ml}{\delta^2}\ln{\frac{\zeta(p_1,p_2)-\mathrm{e}^{-\Omega(p_2)t}}{\zeta(p_1,p_2)-1}}\right\}}, \end{matrix}$

(3.24)

其中 $\Omega(p_2)=\sqrt{m^2-2\delta^2p_2}$ , 且

$\begin{matrix} \hat{c}(p_1,p_2) = \frac{m - \Omega(p_2)}{\delta^2} - \frac{2\Omega(p_2)} {\delta^2[\zeta(p_1,p_2)\mathrm{e}^{\Omega(p_2)t} - 1]},\ \zeta(p_1,p_2) = 1 - \frac{2\Omega(p_2)}{\Omega(p_2) - m + p_1\delta^2}. \end{matrix}$

(3.25)

事实上, 由引理 3.1 知 $m^2>-2\delta^2(a+b)$ , 于是由 (3.24) 式有 $E\left\{\mathrm{e}^{-C_2(u,u+v)r_u-(a+b)\xi(u)}\right\}$ , 从而得到 (3.23) 式. 引理得证.

定理3.1 考虑二维带扰动保险风险模型 (1.1). 设随机向量 $(X,Y)$ 的边际分布为 $F\in\mathcal{R}_{-\alpha}$ 和 $G\in\mathcal{R}_{-\beta}$ , $\alpha, \beta>0$ . 设随机向量 $(X,Y)$ 服从二元 Sarmanov 分布, 即 (2.6) 式. 设对数投资收益过程 $\{\xi(t),t\geq 0\}$ 由 (3.1) 和 (3.2) 式构造, 且 $\{\xi(t),t\geq 0\}$ , $\{(X_i,Y_i),\ i\geq1 \}$ , $\{\theta_i,i\geq1\}$ 和 $\{(B_1(t),B_2(t)),t\geq0\}$ 独立. 于是, 对任意的 $t>0$ 有

$\begin{align*} \Psi(x,y;t) \sim\bar{F}(x)\bar{G}(y) &\left\{ \lambda\widetilde{C}_d\int_{0}^{t}E\mathrm{e}^{-(\alpha+\beta)\xi(u)}\mathrm{d}u + \lambda^2\iint_{u+v\leq t}E\mathrm{e}^{-\beta\xi(u)-\alpha\xi(u+v)}\right.\\&\left. +E\mathrm{e}^{-\alpha\xi(u)-\beta\xi(u+v)}\mathrm{d}u\mathrm{d}v \right\}, \end{align*}$

其中 $E\mathrm{e}^{-(\alpha+\beta)\xi(u)}$ , $E\mathrm{e}^{-\beta\xi(u)-\alpha\xi(u+v)}$ 和 $E\mathrm{e}^{-\alpha\xi(u)-\beta\xi(u+v)}$ 可由引理 3.1 得到.

证本定理的证明分成两部分. 首先证明 Vasicek 模型情形. 由 (3.3) 式有

$\begin{align*} E\mathrm{e}^{-\kappa\xi(t)} =\exp{\left\{\widetilde{A}_1(\kappa)+ \widetilde{A}_2(\kappa)t+\widetilde{A}_3(\kappa)\mathrm{e}^{-mt}+ \widetilde{A}_4(\kappa)\mathrm{e}^{-2mt}\right\}} \end{align*}$

其中 $\widetilde{A}_1(\kappa)=\frac{\kappa}{m}(l-r_0)-3\frac{\kappa^2\delta^2}{4m^3}$ , $\widetilde{A}_2(\kappa)=\kappa\left(\frac{\kappa\delta^2}{2m^2}-l\right)$ , $\widetilde{A}_3(\kappa)=\frac{\kappa}{m}\left(r_0-l+\frac{\kappa\delta^2}{m^2}\right)$ , $\widetilde{A}_4(\kappa)=-\frac{\kappa^2\delta^2}{4m^3}$ . 则 $\int_0^tE\mathrm{e}^{-\kappa\xi(s)}\mathrm{d}s<\infty$ 对任意的 $\kappa>0$ 成立, 于是由定理 (2.1) 即可证得结果.

其次证明 CIR 模型情形. 由 (3.4) 式得

$\begin{gather*} E\mathrm{e}^{-\kappa\xi(t)} =\Big(\frac{\mathrm{e}^{mt/2}}{ \cosh(k_{\kappa}t) +\frac{m}{2k_{\kappa}}\sinh(k_{\kappa}t) }\Big)^{2ml/\delta^2} \cdot\exp\Big\{ -\frac{2\kappa r_0}{m+2k_{\kappa}\coth{(k_{\kappa}t)}} \Big\}. \end{gather*}$

则 $\int_0^tE\mathrm{e}^{-\kappa\xi(s)}\mathrm{d}s<\infty$ 对任意的 $\kappa>0$ 成立, 于是由定理 (2.1) 得到结果. 定理 3.1 得证.

3.2 应用于 Heston 模型

1993 年, Steven Heston 建立了一类随机波动率模型, Heston 模型^[14]. 与 Black-Scholes 模型相比, Heston 模型中波动率是任意的. 令投资收益过程 $\left\{S_t=\mathrm{e}^{\xi(t)},\ t\geq0 \right\}$ 由 Heston 模型描述, 即

$\begin{equation} \mathrm{d}S_t =\mu S_t\mathrm{d}t+\sqrt{r_t}S_t\mathrm{d}W_t^{(1)}, \end{equation}$

(3.26)

其中, $\mu$ 是无风险利率; $\{W_t^{(1)}, t\geq0\}$ 是标准布朗运动; $\{r_t, t\geq0\}$ 是方差过程且

$\begin{equation} \mathrm{d}r_t =m(l-r_t)\mathrm{d}t+\delta\sqrt{r_t}\mathrm{d}W_t^{(2)}. \end{equation}$

(3.27)

此处, $r_0>0$ ; $l>0$ 表示长期均值水平; $m>0$ 表示恢复到均值水平 $l$ 的速率; $\delta>0$ 反映了方差过程波动, 且满足 $2ml>\delta^2$ ; $\{W_t^{(2)}, t\geq0\}$ 是标准布朗运动且与布朗运动 $\{W_t^{(1)}, t\geq0\}$ 相关, 即

$\begin{equation} \mathrm{d}W_t^{(1)} =\tilde{\rho} \mathrm{d}W_t^{(2)} +\sqrt{1-\tilde{\rho}^2}\mathrm{d}W_t^{(3)}, \end{equation}$

(3.28)

其中, $\tilde{\rho}\in[-1,0]$ 是相关系数, $\{W_t^{(3)}, t\geq0\}$ 是标准布朗运动且与 $\{W_t^{(2)}, t\geq0\}$ 独立.

引理3.2 设投资收益过程 $\{S_t=\mathrm{e}^{\xi(t)},t\geq 0\}$ 由 (3.26)-(3.28) 式描述. 对实数 $a, b>0$ , 若

$\begin{equation} (\delta^2+2m\delta)(a+b)+(a+b)^2\delta^2<m^2, \end{equation}$

(3.29)

则

$\begin{matrix} E\mathrm{e}^{-b\xi(u)-a\xi(u+v)} &=\exp\left\{ -b\mu u-a\mu(u+v)+\frac{a\tilde{\rho} mlv}{\delta} \right.\nonumber\\ &\left.\qquad\qquad +\frac{(a+b)\tilde{\rho} (r_0+mlu)}{\delta}+f\,(u,v)+g(u) \right\}, \end{matrix}$

(3.30)

其中 $f(u,v) =\frac{(m-\Omega(e_2))mlv}{\delta^2} -\frac{2ml}{\delta^2}\ln\left(\frac{\zeta(e_1,e_2)-\mathrm{e}^{(u-v)\Omega(e_2)}}{\zeta(e_1,e_2)-\mathrm{e}^{u\Omega(e_2)}}\right)$ , $e_1=-\frac{a\tilde{\rho}}{\delta}$ , $e_2=\frac{a+a^2(1-\tilde{\rho}^2)}{2} -\frac{am\tilde{\rho}}{\delta}$ , $e_3=-\frac{b\tilde{\rho}}{\delta}+\hat{c}(e_1,e_2)$ , $e_4=\frac{(b+a)+(b+a)^2(1-\tilde{\rho}^2)}{2} -\frac{(b+a)m\tilde{\rho}}{\delta}$ , $g(u)=\hat{c}(e_3,e_4)r_0 +\frac{(m-\Omega(e_4))mlu}{\delta^2} -\frac{2ml}{\delta^2}\ln\left(\frac{\zeta(e_3,e_4)-\mathrm{e}^{-\Omega(e_4)u}}{\zeta(e_3,e_4)-1}\right)$ , 且 $\hat{c}(\cdot,\cdot)$ , $\Omega(\cdot)$ 和 $\zeta(\cdot,\cdot)$ 由 (3.25) 式引入.

证由 Itô 公式, (3.26) 式的解为

$\begin{matrix} \mathrm{e}^{\xi(t)}=S_t&=\exp\left\{\mu t-\int_0^t \frac{r_s}{2} \mathrm{d}s +\int_0^t \sqrt{r_s}\,\mathrm{d}W_s^{(1)}\right\}\nonumber\\ &=\exp\left\{\mu t-\int_0^t \frac{r_s}{2} \mathrm{d}s +\tilde{\rho}\int_0^t \sqrt{r_s}\mathrm{d}W_s^{(2)}+\sqrt{1-\tilde{\rho}^2}\int_0^t \sqrt{r_s}\,\mathrm{d}W_s^{(3)}\right\}. \end{matrix}$

(3.31)

将 (3.31) 式代入 $E\mathrm{e}^{-b\xi(u)-a\xi(u+v)}$ 得

$\begin{matrix} &\quad\, E\mathrm{e}^{-b\xi(u)-a\xi(u+v)}=E\left(S_u^{-b}S_{u+v}^{-a}\right)\nonumber\\ &=\mathrm{e}^{-b\mu u-a\mu(u+v)}E\left[\exp\left( \frac{b}{2}\int_0^u r_s\, \mathrm{d}s -b\tilde{\rho}\int_0^u \sqrt{r_s}\,\mathrm{d}W_s^{(2)}-b\sqrt{1-\tilde{\rho}^2}\int_0^u \sqrt{r_s}\, \mathrm{d}W_s^{(3)}\right)\right.\nonumber\\ &\left.\quad \cdot\exp\left( \frac{a}{2}\int_0^{u+v} r_s \,\mathrm{d}s -a\tilde{\rho}\int_0^{u+v} \sqrt{r_s}\,\mathrm{d}W_s^{(2)}-a\sqrt{1-\tilde{\rho}^2}\int_0^{u+v} \sqrt{r_s}\,\mathrm{d}W_s^{(3)}\right)\right]\nonumber\\ &=\mathrm{e}^{-b\mu u-a\mu(u+v)}E\left\{ \exp\left( \frac{b+a}{2}\int_0^u r_s \,\mathrm{d}s -(b+a)\tilde{\rho}\int_0^u \sqrt{r_s}\,\mathrm{d}W_s^{(2)}\right.\right.\nonumber\\ &\quad\left.\left. -(b+a)\sqrt{1-\tilde{\rho}^2}\int_0^u \sqrt{r_s}\,\mathrm{d}W_s^{(3)}\right)\right.\nonumber\\ &\left.\quad \cdot E\left[\exp\left( \frac{a}{2}\int_u^{u+v} r_s \,\mathrm{d}s -a\tilde{\rho}\int_u^{u+v} \sqrt{r_s}\,\mathrm{d}W_s^{(2)}-a\sqrt{1-\tilde{\rho}^2}\int_u^{u+v} \sqrt{r_s}\,\mathrm{d}W_s^{(3)} \right) |\mathcal{F}_u\right]\nonumber \right\}\\ &=:\mathrm{e}^{-b\mu u-a\mu(u+v)}E[M_1(u)\cdot M_2(u,v)]. \end{matrix}$

(3.32)

对 $M_2(u,v)$ , 由 Girsanov 定理,

$\begin{matrix} M_2(u,v) &=E\left\{E\left[\exp\left( \frac{a}{2}\int_u^{u+v} r_s \,\mathrm{d}s -a\tilde{\rho}\int_u^{u+v} \sqrt{r_s}\,\mathrm{d}W_s^{(2)}-a\sqrt{1-\tilde{\rho}^2}\int_u^{u+v} \sqrt{r_s}\,\mathrm{d}W_s^{(3)} \right) \right.\right.\nonumber\\ &\quad\ \left.\left. \cdot \mid\mathcal{F}_u,W_s^{(2)},u\leq s\leq u+v\right]\mid \mathcal{F}_u \right\}\nonumber\\ &=E\left\{ \exp\left( \frac{a}{2}\int_{u}^{u+v} r_s \,\mathrm{d}s -a\tilde{\rho}\int_{u}^{u+v} \sqrt{r_s}\,\mathrm{d}W_s^{(2)} \right)\right. \nonumber\\ &\quad\ \cdot E\left[\exp\left( -a\sqrt{1-\tilde{\rho}^2}\int_{u}^{u+v} \sqrt{r_s}\,\mathrm{d}W_s^{(3)} \right.\right.\left.\left. \mid \mathcal{F}_u,W_s^{(2)},u\leq s\leq u+v\right] \mid \mathcal{F}_u\right\}\nonumber\\ &= E\left[ \exp\left( \frac{a}{2}\int_{u}^{u+v} r_s \,\mathrm{d}s -a\tilde{\rho}\int_{u}^{u+v} \sqrt{r_s}\,\mathrm{d}W_s^{(2)} +\frac{a^2(1-\tilde{\rho}^2)}{2}\int_{u}^{u+v} r_s \,\mathrm{d}s \right)\mid \mathcal{F}_u \right]. \end{matrix}$

(3.33)

更进一步地, 由 (3.27), (3.29) 和 (3.24) 式得

$\begin{matrix} M_2(u,v) &=E\left\{ \exp\left[ \frac{a+a^2(1-\tilde{\rho}^2)}{2}\int_{u}^{u+v} r_s \,\mathrm{d}s \right.\right. \\ &\quad\ \left.\left. -\frac{a\tilde{\rho}}{\delta}\left( m\int_{u}^{u+v} r_s \,\mathrm{d}s-mlv+r_{u+v}-r_{u} \right)\right]\mid \mathcal{F}_u \right\}\nonumber\\ &=\exp\left(\frac{a\tilde{\rho} mlv+a\tilde{\rho} r_{u}}{\delta}\right) \\ &\quad\ \cdot E\left\{ \exp\left[ -\frac{a\tilde{\rho}}{\delta}r_{u+v} +\left(\frac{a+a^2(1-\tilde{\rho}^2)}{2}-\frac{am\tilde{\rho}}{\delta}\right)\int_u^{u+v} r_s \,\mathrm{d}s \right] |\mathcal{F}_u\right\}\nonumber\\ &=\exp\left[ \frac{a\tilde{\rho} mlv+a\tilde{\rho} r_u}{\delta} +\hat{c}\left(-\frac{a\tilde{\rho}}{\delta}, \frac{a+a^2(1-\tilde{\rho}^2)}{2}-\frac{am\tilde{\rho}}{\delta}\right)r_u +f\,(u,v)\right], \end{matrix}$

(3.34)

其中 $f\,(u,v)$ 由引理 3.2 引入, $\hat{c}(\cdot,\cdot)$ 由 (3.25) 式引入. 故

$\begin{align*} E\mathrm{e}^{-b\xi(u)-a\xi(u+v)} &=\exp\{-b\mu u-a\mu(u+v)\}E\left\{M_1(u) \exp\left\{\frac{ a\tilde{\rho} mlv+a\tilde{\rho} r_u }{\delta} \right.\right.\\ &\quad\left.\left. +\hat{c}\left(-\frac{a\tilde{\rho}}{\delta}, \frac{a+a^2(1-\tilde{\rho}^2)}{2}-\frac{am\tilde{\rho}}{\delta}\right)r_u+f\,(u,v) \right\} \right\}\nonumber\\ &=\exp\{-b\mu u-a\mu(u+v)+\frac{a\tilde{\rho} mlv}{\delta} +f\,(u,v)\}\\ &\quad\cdot E\left( \exp\left\{ \left[\frac{a\tilde{\rho}}{\delta}+\hat{c}\left(-\frac{a\tilde{\rho}}{\delta}, \frac{a+a^2(1-\tilde{\rho}^2)}{2}-\frac{am\tilde{\rho}}{\delta} \right)\right]r_u+\frac{b+a}{2}\int_0^u r_s \,\mathrm{d}s \right.\right.\nonumber\\ &\left.\left.\quad -(b+a)\tilde{\rho}\int_0^u \sqrt{r_s}\,\mathrm{d}W_s^{(2)} -(b+a)\sqrt{1-\tilde{\rho}^2}\int_0^u \sqrt{r_s}\,\mathrm{d}W_s^{(3)} \right\} \right)\\ &:=\exp\{-b\mu u-a\mu(u+v)+a\tilde{\rho} mlv/\delta+f(u,v)\}\widetilde{M}_1(u).\nonumber \end{align*}$

对 $\widetilde{M}_1(u)$ , 采用类似于 (3.33) 和 (3.34) 式的推导过程即可得到 (3.30) 式. 证毕.

定理3.2 考虑二维带扰动的保险风险模型 (1.1). 设随机向量 $(X,Y)$ 的边际分布为 $F\in\mathcal{R}_{-\alpha}$ 和 $G\in\mathcal{R}_{-\beta}$ , $\alpha, \beta>0$ , 而且 $(X,Y)$ 服从二元 Sarmanov 分布, 即 (2.6) 式. 设投资收益过程 $\{S_t=\mathrm{e}^{\xi(t)},\ t\geq0 \}$ 由 (3.26)-(3.28) 式构造, 且 $\{\xi(t),t\geq 0\}$ , $\{(X_i,Y_i),i\geq1\}$ , $\{\theta_i,i\geq1\}$ 和 $\{(B_1(t),B_2(t)),t\geq0\}$ 独立. 若存在常数 $\kappa>\max{\{\alpha+\beta,2\}}$ 满足 $\delta\kappa[2m+(\kappa+1)\delta]<m^2$ , 则对任意的 $t>0$ ,

$\begin{align*} \Psi(x,y;t) \sim\bar{F}(x)\bar{G}(y) &\bigg\{ \lambda\widetilde{C}_d\int_{0}^{t}E\mathrm{e}^{-(\alpha+\beta)\xi(u)}\,\mathrm{d}u +\lambda^2\iint_{u+v\leq t}E\mathrm{e}^{-\beta\xi(u)-\alpha\xi(u+v)}\\ &\ +E\mathrm{e}^{-\alpha\xi(u)-\beta\xi(u+v)}\,\mathrm{d}u\mathrm{d}v \bigg\}, \end{align*}$

其中 $E\mathrm{e}^{-(\alpha+\beta)\xi(u)}$ , $E\mathrm{e}^{-\alpha\xi(u)-\beta\xi(u+v)}$ 和 $E\mathrm{e}^{-\beta\xi(u)-\alpha\xi(u+v)}$ 可根据 (3.30) 式得到.

证由引理 3.2, 在 (3.32)-(3.34) 式的推导过程中, 令 $b=0$ , $a=\kappa$ , $u=0$ 及 $v=t$ 得

$\begin{align*} E\big\{\mathrm{e}^{-\kappa\xi(t)}\big\} &=\exp\left\{-\kappa\mu t +\frac{\kappa\tilde{\rho} mlt}{\delta} +\frac{\kappa\tilde{\rho} r_0}{\delta} +\hat{c}\left(-\frac{\kappa\tilde{\rho}}{\delta}, \frac{\kappa+\kappa^2(1-\tilde{\rho}^2)}{2} -\frac{\kappa m\tilde{\rho}}{\delta} \right)+f\,(0,t)\right\}\\ &<\infty, \end{align*}$

于是根据定理 (2.1) 即可得到结果, 定理 (3.2) 得证.

4 主要结果的证明

4.1 一些引理

首先引入若干证明主要结果需要的关系式. 设 $F$ 是定义于 $[0,\infty)$ 的分布函数, 若 $F\in\mathcal{R}_{-\alpha}$ , $0<\alpha<\infty$ , 则对 $0<a<b<\infty$ , 下式对任意的 $y\in[a,b]$ 一致成立,

$\begin{matrix} \bar{F}(xy)\sim y^{-\alpha}\bar{F}(x). \end{matrix}$

(4.1)

根据专著 ^[2], 若 $F\in\mathcal{R}_{-\alpha}$ , $\alpha>0$ , 则存在常数 $C_F>1$ 和 $D_F>0$ , 使得对任意的 $0<\delta_0<\alpha$ 和 $x, xy\geq D_F$ 有

$\begin{matrix} \frac{1}{C_F}(y^{-\alpha+\delta_0}\wedge y^{-\alpha-\delta_0}) \leq \frac{\bar{F}(xy)}{\bar{F}(x)} \leq C_F(y^{-\alpha+\delta_0}\vee y^{-\alpha-\delta_0}). \end{matrix}$

(4.2)

于是对任意的 $p>\alpha$ , 由(4.2)式得

$\begin{equation} x^{-p}=o(\bar{F}(x)),\quad x\rightarrow \infty. \end{equation}$

(4.3)

设 $X$ 和 $Y$ 是两个独立的随机变量, 若 $X$ 的分布函数 $F\in\mathcal{R}_{-\alpha}$ , 且对 $p>\alpha$ 非负随机变量 $Y$ 满足 $EY^{p}<\infty$ , 则由文献 [24] 的 (4.4) 式, 对任意给定的 $M\geq0$ 有

$\begin{matrix} \lim\limits_{x\rightarrow\infty} \frac{P\left( XY>x, Y>M\right)}{\bar{F}(x)} =E\left( Y^{\alpha}\mathbb{I}_{[Y>M]} \right), \end{matrix}$

(4.4)

同时, 根据文献 [15] 得到对任意固定的 $\delta>0$ , $\alpha<p<\infty$ 和足够大的 $x$ , 存在与 $Y$ 和 $\delta$ 无关的常数 $C>0$ 使得下式成立

$\begin{matrix} P\left( XY>\delta x\,\big|\, Y \right) \leq C\bar{F}(x) \left(\delta^{-p}Y\,^{p} \mathbb{I}_{[Y\geq\delta]} + \mathbb{I}_{[Y<\delta]}\right). \end{matrix}$

(4.5)

引理4.1 设 $X^{\,*}$ , $Y^{\,*}$ 和 $X^{\,**}$ 是非负独立的随机变量, 其分布函数分别为 $F\in\mathcal{R}_{-\alpha}$ , $G\in\mathcal{R}_{-\beta}$ 和 $F^{\,**}\in\mathcal{R}_{-\alpha}$ , $\alpha>0$ , $\beta>0$ . 设非负随机变量 $\Theta_1$ 和 $\Theta_2$ 是非退化的且与 $X^{\,*}$ , $Y^{\,*}$ 和 $X^{\,**}$ 独立. 若存在常数 $\eta>\alpha+\beta$ 满足

$\begin{matrix} E\Theta_k^{\eta}<\infty,\qquad k=1,2, \end{matrix}$

(4.6)

和 $P\left\{\omega:\Theta_1(\omega)\Theta_2(\omega)\neq0\right\}>0$ , 则

$\begin{matrix} &\,\,P\left\{ X^{\,*}\Theta_1>x,\ Y^{\,*}\Theta_2>y \right\} \sim \bar{F}(x)\bar{G}(y)E\left( \Theta_1^{\alpha}\Theta_2^{\beta} \right), \end{matrix}$

(4.7)

$\begin{matrix} P\left\{ X^{\,**}\Theta_1>x,\ Y^{\,*}\Theta_1>y,\ X^{\,*}\Theta_2>x \right\}=o(1)\bar{F}(x)\bar{G}(y). \end{matrix}$

(4.8)

证由 (4.6) 式和 Hölder 不等式, 对任意的 $0<l\leq\eta$ 有

$\begin{matrix} E\Theta_k^{l} \leq \left(E\Theta_k^{\eta}\right)^{l/\eta} <\infty,\quad k=1,2. \end{matrix}$

(4.9)

对任意的 $\varepsilon>0$ , 取 $b\in (1,\infty)$ 和 $a\in(0,1)$ 满足 $a^{\alpha} \vee a^{\beta}<\varepsilon$ , $\alpha+\beta+2\varepsilon<\eta$ 和

$\begin{matrix} E\left\{ \Theta_k^{l} \left(\mathbb{I}_{[\Theta_k<a]} +\mathbb{I}_{[\Theta_k>b]} \right) \right\} <\varepsilon. \end{matrix}$

(4.10)

首先证明 (4.7) 式. 先证明概率 $P\left\{ X^{\,*}\Theta_1>x,\ Y^{\,*}\Theta_2>y \right\}$ 的渐近上界, 令 $E_{1}=\left\{ \Theta_1<a \right\}$ , $E_{2}=\left\{ a\leq\Theta_1\leq b \right\}$ , $E_{3}=\left\{ \Theta_1>b \right\}$ . 于是 $P\left\{ X^{\,*}\Theta_1>x,\ Y^{\,*}\Theta_2>y \right\}$ 可重写为 $P\{ X^{\,*}\Theta_1>x,$ $Y^{\,*}\Theta_2>y \} = \sum\limits_{s=1}^3P\left\{ X^{\,*}\Theta_1>x, Y^{\,*}\Theta_2>y,E_{s}\right\} :=\sum\limits_{s=1}^3 Q_{s}$ . 由 $F\in\mathcal{R}_{-\alpha}$ , $G\in\mathcal{R}_{-\beta}$ , (4.4) 及 (4.9) 式知, 存在 $x_1>0$ , $y_1>0$ 使得当 $x>x_1$ , $y>y_1$ 时,

$\begin{align*} Q_{1} \leq \bar{F}\left( \frac{x}{a} \right) P\left\{ Y^{\,*}\Theta_2>y \right\} \sim a^{\alpha}\bar{F}(x)\bar{G}(y)E\Theta_2^{\beta} <C\varepsilon\bar{F}(x)\bar{G}(y). \end{align*}$

根据 (4.4), (4.1), (4.10) 和 (4.9) 式知, 存在 $x_2>x_1$ , $y_2>y_1$ 使得当 $x>x_2$ , $y>y_2$ 时,

$\begin{align*} Q_{2} &= P\left\{ X^{\,*}\Theta_1>x, Y^{\,*}\Theta_2>y,a\leq\Theta_1\leq b, \Theta_2<a \right\} +P\left\{ X^{\,*}\Theta_1>x, Y^{\,*}\Theta_2>y, \right.\\ &\left.\quad\ a\leq\Theta_1\leq b, a\leq\Theta_2\leq b \right\} +P\left\{ X^{\,*}\Theta_1>x, Y^{\,*}\Theta_2>y,a\leq\Theta_1\leq b, \Theta_2>b \right\}\\ &\leq \bar{G}\left(\frac{y}{a}\right)P\left\{ X^{\,*}\Theta_1>x \right\} +\int_a^b\int_a^b \bar{F}\left(x/u\right)\bar{G}\left(y/v\right)P\left(\Theta_1\in \mathrm{d}u,\Theta_2 \in \mathrm{d}v \right)\\ &\quad\ +\bar{F}\left( \frac{x}{b} \right)P\left\{ Y^{\,*}\Theta_2>y, \Theta_2>b \right\}\\ &\sim a^{\beta}\bar{F}(x)\bar{G}(y)E\Theta_1^{\alpha} +\bar{F}(x)\bar{G}(y) \int_a^b\int_a^b u^{\alpha}v^{\beta}P\left(\Theta_1\in \mathrm{d}u,\Theta_2\in \mathrm{d}v \right) \\ &\quad\ +b^{\alpha}\bar{F}(x)\bar{G}(y)E\left(\Theta_2^{\beta}\mathbb{I}_{[\Theta_2>b]}\right)\\ &\leq C\varepsilon\bar{F}(x)\bar{G}(y)+ \bar{F}(x)\bar{G}(y) E\left(\Theta_1^{\alpha}\Theta_2^{\beta}\right). \end{align*}$

由 (4.4), (4.5), (4.10) 式和 Hölder 不等式, 存在 $x_3>x_2$ , $y_3>y_2$ 使得对任意的 $x>x_3$ , $y>y_3$ 有

$\begin{align*} Q_{3} &= P\left\{ X^{\,*}\Theta_1>x,Y^{\,*}\Theta_2>y,\Theta_1> b, \Theta_2\leq b \right\} \!+\!P\left\{ X^{\,*}\Theta_1>x,Y^{\,*}\Theta_2>y,\Theta_1> b, \Theta_2> b \right\}\\ &\leq \bar{G}\left(\frac{y}{b}\right) P\left\{ X^{\,*}\Theta_1>x,\Theta_1> b \right\} \\ &\quad\ +E\left\{\mathbb{I}_{[\Theta_1> b,\Theta_2> b]} P\left(X^{\,*}\Theta_1>x|\Theta_1,\Theta_2 \right) P\left(Y^{\,*}\Theta_2>y|\Theta_1,\Theta_2 \right) \right\}\\ &\leq C\varepsilon\bar{F}(x)\bar{G}(y) \\ &\quad\ +C\bar{F}(x)\bar{G}(y)E\left\{ \mathbb{I}_{[\Theta_1> b,\Theta_2> b]} \left( \Theta_1^{\alpha+\varepsilon}\mathbb{I}_{[\Theta_1\geq 1]}+\mathbb{I}_{[\Theta_1< 1]}\right) \left( \Theta_2^{\beta+\varepsilon}\mathbb{I}_{[\Theta_2\geq 1]}+\mathbb{I}_{[\Theta_2< 1]}\right) \right\}\\ &\leq C\varepsilon\bar{F}(x)\bar{G}(y)+ C\bar{F}(x)\bar{G}(y)E\left\{ \Theta_1^{\alpha+\varepsilon}\mathbb{I}_{[\Theta_1> b]} \Theta_2^{\beta+\varepsilon}\mathbb{I}_{[\Theta_2> b]} \right\}\\ &\leq C\varepsilon\bar{F}(x)\bar{G}(y) + C\bar{F}(x)\bar{G}(y) \left[ E\left( \Theta^{\alpha+\beta+2\varepsilon}_1 \mathbb{I}_{[\Theta_1> b]}\right)\right] ^{\frac{\alpha+\varepsilon}{\alpha+\beta+2\varepsilon}} \left[E\left(\Theta^{\alpha+\beta+2\varepsilon}_2 \mathbb{I}_{[\Theta_2> b]} \right)\right] ^{\frac{\beta+\varepsilon}{\alpha+\beta+2\varepsilon}}\\ &\leq C\varepsilon\bar{F}(x)\bar{G}(y). \end{align*}$

于是, 由上述关于 $Q_1$ , $Q_2$ 和 $Q_3$ 的讨论得到概率 $P\left\{ X^{\,*}\Theta_1>x,\ Y^{\,*}\Theta_2>y \right\}$ 的渐近上界, 现证明概率 $P\left\{ X^{\,*}\Theta_1>x,\ Y^{\,*}\Theta_2>y \right\}$ 的渐近下界. 根据 $Q_2$ 的讨论, 存在 $x_4>x_3$ , $y_4>y_3$ 使得当 $x>x_4$ , $y>y_4$ 时,

$\begin{align*} P\left\{ X^{\,*}\Theta_1>x,\ Y^{\,*}\Theta_2>y \right\} &\geq P\left\{ X^{\,*}\Theta_1>x, Y^{\,*}\Theta_2>y,a\leq\Theta_1\leq b, a\leq\Theta_2\leq b \right\}\\ &\sim \bar{F}(x)\bar{G}(y)E\left( \Theta_1^{\alpha}\Theta_2^{\beta} \mathbb{I}_{[a\leq\Theta_1\leq b]}\mathbb{I}_{[a\leq\Theta_2\leq b]}\right) \\ & \geq\bar{F}(x)\bar{G}(y) \left\{E\left(\Theta^{\alpha}_1\Theta^{\beta}_2\right) -E\left(\Theta^{\alpha}_1\Theta^{\beta}_2\mathbb{I}_{[\Theta_1<a]} \right)\right.\\ &\left.\quad -E\left(\Theta^{\alpha}_1\Theta^{\beta}_2\mathbb{I}_{[\Theta_1>b]} \right) -E\left(\Theta^{\alpha}_1\Theta^{\beta}_2\mathbb{I}_{[\Theta_2<a]} \right) -E\left(\Theta^{\alpha}_1\Theta^{\beta}_2\mathbb{I}_{[\Theta_2>b]} \right)\right\}. \end{align*}$

由 Hölder 不等式, (4.9) 和 (4.10) 式得

$\begin{align*} E\left(\Theta^{\alpha}_1\Theta^{\beta}_2\mathbb{I}_{[\Theta_1<a]} \right) \leq \left[E\left(\Theta^{\alpha+\beta}_1\mathbb{I}_{[\Theta_1<a]}\right)\right]^{\alpha/(\alpha+\beta)} \left[E\left(\Theta^{\alpha+\beta}_2 \right)\right]^{\beta/(\alpha+\beta)} \leq C\varepsilon. \end{align*}$

类似地, $E\left(\Theta^{\alpha}_1\Theta^{\beta}_2\mathbb{I}_{[\Theta_1>b]} \right) +E\left(\Theta^{\alpha}_1\Theta^{\beta}_2\mathbb{I}_{[\Theta_2<a]} \right) +E\left(\Theta^{\alpha}_1\Theta^{\beta}_2\mathbb{I}_{[\Theta_2>b]}\right) \leq C\varepsilon.$ 于是, 对上述 $\varepsilon>0$ , $x>x_4$ 和 $y>y_4$ ,

$\begin{align*} (1-C\varepsilon)\bar{F}(x)\bar{G}(y)E\left\{\Theta^{\alpha}_1\Theta^{\beta}_2\right\} \leq P\left\{ X^{\,*}\Theta_1>x,\ Y^{\,*}\Theta_2>y \right\} \leq (1+C\varepsilon)\bar{F}(x)\bar{G}(y)E\left\{\Theta^{\alpha}_1\Theta^{\beta}_2\right\}, \end{align*}$

即证明渐近式 (4.7).

其次证明 (4.8) 式. 对上述 $\varepsilon>0$ , 由 (4.2) 式, Hölder 不等式, (4.9) 和 (4.3) 式,

$\begin{align*} &P\left\{ X^{\,**}\Theta_1>x,\ Y^{\,*}\Theta_1>y,\ X^{\,*}\Theta_2>x \right\}\\ =&\left(\int_0^{x/D_F}\int_0^{y/D_G}+\int_0^{x/D_F}\int_{y/D_G}^{\infty} +\int_{x/D_F}^{\infty}\int_0^{y/D_G}+\int_{x/D_F}^{\infty}\int_{y/D_G}^{\infty}\right)\\ &\cdot \bar{F} \left(\frac{x}{v}\right) \bar{G} \left(\frac{y}{u}\right) \bar{F}^{\,**} \left(\frac{x}{u}\right) P\left\{ \Theta_1\in \mathrm{d}u, \Theta_2\in \mathrm{d}v \right\}\\ \leq\ & C\bar{F}(x)\bar{G}(y)\int_0^{x/D_F}\int_0^{y/D_G} (u^{\beta-\varepsilon}\vee u^{\beta+\varepsilon}) (v^{\alpha-\varepsilon}\vee v^{\alpha+\varepsilon}) \bar{F}^{\,**} \left(\frac{x}{u}\right) P\left\{ \Theta_1\in \mathrm{d}u, \Theta_2\in \mathrm{d}v \right\}\\ & +C\bar{F}(x)\int_0^{x/D_F}\int_{y/D_G}^{\infty} (v^{\alpha-\varepsilon}\vee v^{\alpha+\varepsilon}) \left( \frac{uD_G}{y} \right)^{\beta+\varepsilon} P\left\{ \Theta_1\in \mathrm{d}u, \Theta_2\in \mathrm{d}v \right\}\\ & +C\bar{G}(y)\int_{x/D_F}^{\infty}\int_0^{y/D_G} (u^{\beta-\varepsilon}\vee u^{\beta+\varepsilon}) \left( \frac{vD_F}{x} \right)^{\alpha+\varepsilon} P\left\{ \Theta_1\in \mathrm{d}u, \Theta_2\in \mathrm{d}v \right\}\\ & +C\int_{x/D_F}^{\infty}\int_{y/D_G}^{\infty} \left( \frac{uD_G}{y} \right)^{\beta+\varepsilon} \left( \frac{vD_F}{x} \right)^{\alpha+\varepsilon} P\left\{ \Theta_1\in \mathrm{d}u, \Theta_2\in \mathrm{d}v \right\} \\ \leq\ & C\bar{F}(x)\bar{G}(y)E\left\{ (\Theta_1^{\beta-\varepsilon}+\Theta_1^{\beta+\varepsilon}) (\Theta_2^{\alpha-\varepsilon}+\Theta_2^{\alpha+\varepsilon}) \mathbb{I}_{[X^{\,**}\Theta_1>x]} \right\} \\ & +C\bar{F}(x)y^{-(\beta+\varepsilon)}E\left\{ (\Theta_2^{\alpha-\varepsilon}+ \Theta_2^{\alpha+\varepsilon}) \Theta_1^{\beta+\varepsilon}\right\} +Cx^{-(\alpha+\varepsilon)}\bar{G}(y)E\left\{ (\Theta_1^{\beta-\varepsilon}+ \Theta_1^{\beta+\varepsilon}) \Theta_2^{\alpha+\varepsilon} \right\}\\ & +Cx^{-(\alpha+\varepsilon)}y^{-(\beta+\varepsilon)}E\left\{ \Theta_1^{\beta+\varepsilon}\Theta_2^{\alpha+\varepsilon} \right\} =o(1)\bar{F}(x)\bar{G}(y). \end{align*}$

引理得证.

设 $\left\{(\widetilde{X}^{\,*}_k, \widetilde{Y}^{\,*}_k,\widetilde{\theta}^{\,*}_k),k\geq 1\right\}$ 是独立同分布的随机序列, 其独立复制为 $(\widetilde{X}^{\,*}, \widetilde{Y}^{\,*},\widetilde{\theta}^{\,*})$ . 设 $(\widetilde{X}^{\,*},$ $\widetilde{Y}^{\,*},\widetilde{\theta}^{\,*})$ 与本文中的随机量独立且边际分布分别为 $\widetilde{F}$ , $\widetilde{G}$ 和 $\widetilde{H}$ , 即

$\begin{gather*} \mathrm{d}\widetilde{F}(x)=\left( 1-\frac{\varphi_1(x)}{b_1}\right)\mathrm{d}F(x),\; \mathrm{d}\widetilde{G}(y)=\left( 1-\frac{\varphi_2(y)}{b_2}\right)\mathrm{d}G(y),\; \mathrm{d}\widetilde{H}(z)=\left( 1-\frac{\varphi_3(z)}{b_3} \right)\mathrm{d}H(z), \end{gather*}$

其中, $\varphi_1(x)$ , $\varphi_2(y)$ 和 $\varphi_3(z)$ 由假设 2.1 引入, $b_i$ , $i=1,2,3$ , 由 (2.9) 式引入. 于是,

$\begin{matrix} \bar{\widetilde{F}}(x)=\int_x^{\infty} \left(1-\varphi_1(u)/b_1\right)\mathrm{d}F(u) \sim \left(1-d_1/b_1 \right)\bar{F}(x),\quad x\rightarrow\infty, \end{matrix}$

(4.11)

$\begin{matrix} \bar{\widetilde{G}}(y)=\int_y^{\infty} \left(1-\varphi_2(v)/b_2\right)\mathrm{d}G(v) \sim \left(1-d_2/b_2\right)\bar{G}(y),\quad y\rightarrow\infty, \end{matrix}$

(4.12)

其中 $d_1$ 和 $d_2$ 由 (2.4) 式引入. 进一步地, 对 $0<\alpha, \beta<\infty$ , 由 $F\in\mathcal{R}_{-\alpha}$ 和 $G\in\mathcal{R}_{-\beta}$ 知 $\widetilde{F} \in\mathcal{R}_{-\alpha}$ 和 $\widetilde{G}\in\mathcal{R}_{-\beta}$ .

引理4.2 在假设 2.1 成立的条件下,

$\begin{align*} P\left\{g(X,Y,\theta)\in\Delta\right\} =\ &f_0P\left\{g(X^{\,*},Y^{\,*},\theta^*)\in\Delta\right\}-(f_{12}+f_{13})P\left\{g(\widetilde{X}^{\,*},Y^{\,*},\theta^*)\in\Delta\right\}\\ &-(f_{12}+f_{23})P\left\{g(X^{\,*},\widetilde{Y}^{\,*},\theta^*)\in\Delta\right\} \\ &-(f_{13}+f_{23})P\left\{g(X^{\,*},Y^{\,*},\widetilde{\theta}^*)\in\Delta\right\} +f_{12}P\left\{g(\widetilde{X}^{\,*},\widetilde{Y}^{\,*},\theta^*) \in\Delta\right\}\\ &+f_{13}P\left\{g(\widetilde{X}^{\,*},Y^{\,*},\widetilde{\theta}^*)\in\Delta\right\} +f_{23}P\left\{g(X^{\,*},\widetilde{Y}^{\,*},\widetilde{\theta}^*)\in\Delta\right\}, \end{align*}$

其中, $g$ 是 Borel 可测函数, $\Delta$ 是 Borel 集, 且 $f_0=1+\eta_{12}b_1b_2+\eta_{13}b_1b_3+\eta_{23}b_2b_3$ , $f_{12}=\eta_{12}b_1b_2$ , $f_{13}=\eta_{13}b_1b_3$ , $f_{23}=\eta_{23}b_2b_3$ .

证由假设 2.1 及 $\widetilde{F}(x)$ , $\widetilde{G}(y)$ 和 $\widetilde{H}(z)$ 的定义得,

$\begin{align*} P\left\{g(X,Y,\theta)\in\Delta\right\} = &\int_0^{\infty}\int_0^{\infty}\int_0^{\infty}P\left\{g(u,v,w)\in\Delta\right\} \left[1+\eta_{12}\varphi_{1}(u)\varphi_{2}(v)+\eta_{13}\varphi_{1}(u)\varphi_{3}(w) \right.\\ &\left. +\eta_{23}\varphi_{2}(v)\varphi_{3}(w)\right] \mathrm{d}F(u)\mathrm{d}G(v)\mathrm{d}H(w)\\ =&\int_0^{\infty}\int_0^{\infty}\int_0^{\infty}P\left\{g(u,v,w)\in\Delta\right\} \left[f_0-(f_{12}+f_{13})\left(1-\varphi_1(u)/b_1\right)\right.\\ & -(f_{12}+f_{23})\left(1-\varphi_2(v)/b_2\right) -(f_{13}+f_{23})\left(1-\varphi_3(v)/b_3\right) \\ & +f_{12}\left(1-\varphi_1(u)/b_1\right)\left(1-\varphi_2(v)/b_2\right) +f_{13}\left(1-\varphi_1(u)/b_1\right)\left(1-\varphi_3(v)/b_3\right) \\ &\left. +f_{23}\left(1-\varphi_2(v)/b_2\right) \left(1-\varphi_3(v)/b_3\right) \right]\mathrm{d}F(u)\mathrm{d}G(v)\mathrm{d}H(w)\\ =\ &f_0P\left\{g(X^{\,*},Y^{\,*},\theta^*)\in\Delta\right\}-(f_{12}+f_{13})P\left\{g(\widetilde{X}^{\,*},Y^{\,*},\theta^*)\in\Delta\right\}\\ & -(f_{12}+f_{23})P\left\{g(X^{\,*},\widetilde{Y}^{\,*},\theta^*)\in\Delta\right\} \\ & -(f_{13}+f_{23})P\left\{g(X^{\,*},Y^{\,*},\widetilde{\theta}^*)\in\Delta\right\} +f_{12}P\left\{g(\widetilde{X}^{\,*},\widetilde{Y}^{\,*},\theta^*)\in\Delta\right\}\\ & +f_{13}P\left\{g(\widetilde{X}^{\,*},Y^{\,*},\widetilde{\theta}^*)\in\Delta\right\} +f_{23}P\left\{g(X^{\,*},\widetilde{Y}^{\,*},\widetilde{\theta}^*)\in\Delta\right\}. \end{align*}$

引理得证.

对任意的 $t>0$ 和整数 $i>0$ , 令 $\tau_i=\theta_1+\theta_2+\cdots+\theta_i$ , $\tau_i^*=\theta_1^*+\theta_2^*+\cdots+\theta_{i}^*$ , $\vartheta_i(t)=\mathrm{e}^{-\xi(\tau_i)}\mathbb{I}_{[\tau_i \leq t]}$ , $\vartheta_i^*(t)=\mathrm{e}^{-\xi(\tau_i^*)}\mathbb{I}_{[\tau_i^* \leq t]}$ , $\widehat{\vartheta}_{ki}(t)=\mathrm{e}^{-\xi(\hat{\tau}^{(k)}_i)}\mathbb{I}_{[\hat{\tau}^{(k)}_i\leq t]}$ , 其中 $k=1,2,3,4,5$ , $\hat{\tau}^{(k)}_i$ 由 (2.13)-(2.15) 式引入. 对任意的 $q \geq 0$ 有

$\begin{matrix} \sum\limits_{n=1}^{\infty} n^{q} \frac{ s^{n-1} \lambda^{n}}{(n-1)!} \mathrm{e}^{-\lambda s} \leq \begin{cases} C(s^q \vee 1), & q=0,1,2\cdots,\\ C(s^{[q]+1} \vee 1), & \text{其它}, \end{cases} \end{matrix}$

(4.13)

其中 $[q]$ 是 $q$ 的整数部分. 对 $q\geq0$ , $0<l \leq \kappa$ 和定理 (2.1) 中的 $\kappa$ , 根据Hölder 不等式,

$\begin{matrix} \sum\limits_{n=1}^{\infty} n^q E\vartheta_n^{\,l}(t) & =\sum\limits_{n=1}^{\infty}n^q E\left\{\mathbb{I}_{[\tau_n \leq t]}E\left(\mathrm{e}^{-l\xi(\tau_n)}|\tau_n \right)\right\} \nonumber\\ & =\sum\limits_{n=1}^{\infty}n^q \int_0^t E(\mathrm{e}^{-l\xi(s)})\frac{s^{n-1}\lambda^n}{(n-1)!}\mathrm{e}^{-\lambda s}{\rm d}s\nonumber\\ &=\int_0^t E(\mathrm{e}^{-l\xi(s)}) \sum\limits_{n=1}^{\infty} n^q \frac{s^{n-1}\lambda^n}{(n-1)!}\mathrm{e}^{-\lambda s}{\rm d}s\nonumber\\ & \leq C(t^{[q]+1} \vee 1)\int_0^t E\mathrm{e}^{-l\xi(s)}{\rm d}s \nonumber\\ & \leq C(T) \int_0^t \left( E\mathrm{e}^{-\kappa\xi(s)} \right)^{l/\kappa}{\rm d}s\nonumber\\ & \leq C(T) \left( \int_0^t E\mathrm{e}^{-\kappa\xi(s)}{\rm d}s \right)^{l/\kappa} <\infty, \quad 0<t\leq T, \end{matrix}$

(4.14)

其中, 上式的第二步是由伽马分布的定义得到. 在 (4.27) 式中取 $q=0$ , 对 $0<l \leq \kappa$ 和 $0<t\leq T$ , 则

$\begin{equation} E\vartheta_n^{\,l}(t)<\infty, \quad \text{对任意的}\ n\geq1. \end{equation}$

(4.15)

引理4.3 在定理 (2.1) 的条件下, 对任意的 $i\geq1$ , $j\geq1$ 和 $0<t\leq T$ , 有

(1) 当 $i=j$ 时, $P\left\{ X_i\vartheta_i(t)>x,\ Y_i\vartheta_i(t)>y \right\}\sim \widetilde{C}_d\bar{F}(x)\bar{G}(y) E\widehat{\vartheta}_{3i}^{\alpha+\beta}(t)$ , 其中 $\widetilde{C}_d$ 由 (2.4) 式引入;

(2) 当 $i<j$ 时, $P\left\{ X_i\vartheta_i(t)>x,\ Y_j\vartheta_j(t)>y \right\}\sim \bar{F}(x)\bar{G}(y)E\left\{ \widehat{\vartheta}^{\alpha}_{1i}(t)\widehat{\vartheta}^{\beta}_{4j}(t) \right\}$ ;

(3) 当 $i>j$ 时, $P\left\{ X_i\vartheta_i(t)>x,\ Y_j\vartheta_j(t)>y \right\}\sim \bar{F}(x)\bar{G}(y) E\left\{\widehat{\vartheta}^{\beta}_{2j}(t)\widehat{\vartheta}^{\alpha}_{5i}(t) \right\}$ .

引入记号

$\begin{equation} \vartheta_i^*(s_k,s_m;t) \nonumber\\ =\mathrm{e}^{-\xi(\theta_1^*+\cdots+\theta_{k-1}^*+s_k+\cdots+\theta_{m-1}^*+s_m+\cdots+\theta_i^*)} \mathbb{I}_{[\theta_1^*+\cdots+s_k+\cdots+s_m+\cdots+\theta_i^*\leq t]}, \ 1\leq k< m \leq i \end{equation}$

(4.16)

和

$\begin{equation} \vartheta_i^*(s_k;t) =\mathrm{e}^{-\xi(\theta_1^*+\cdots+\theta_{k-1}^*+s_k+\cdots+\theta_i^*)}\mathbb{I}_{[\theta_1^*+\cdots+\theta_{k-1}^*+s_k+\cdots+\theta_i^*\leq t]}, \quad 1\leq k \leq i. \end{equation}$

(4.17)

由 (4.15) 式和

$\begin{equation} {\rm d}\widetilde{H}(w) =\left( 1-\varphi_3(v)/b_3 \right){\rm d}H(w), \end{equation}$

(4.18)

得, 对任意的 $n$ 和 $0<l\leq\kappa$ , 有

$\begin{matrix} E\left\{\vartheta_{n}^{*l}(\widetilde{\theta}_n^*;t)\right\} &=E\left\{E\left[\vartheta_{n}^{*l}(\widetilde{\theta}_n^*;t) \mid\widetilde{\theta}_n^*\right]\right\} =\int_0^t E\left(\mathrm{e}^{-l\xi(\theta_1^*+\cdots+u)}\right){\rm d}\widetilde{H}(u)\nonumber\\ &\leq 2\int_0^t E\left(\mathrm{e}^{-l\xi(\theta_1^*+\cdots+u)}\right){\rm d}H(u) \nonumber\\ & =2E\left\{\vartheta_{n}^{*l}(t)\right\} <\infty \end{matrix}$

(4.19)

和

$\begin{equation} E\left\{\vartheta_{n}^{*l}(\widetilde{\theta}_k^*,\widetilde{\theta}_n^*;t)\right\} =E\left\{E\left[\vartheta_{n}^{*l}(\widetilde{\theta}_k^*,\widetilde{\theta}_n^*;t)\mid\widetilde{\theta}_k^*,\widetilde{\theta}_n^*\right]\right\} \leq4E\left\{\vartheta_{n}^{*l}(t)\right\}<\infty. \end{equation}$

(4.20)

对任意的 $i\geq1$ , 证明 $P\left\{ X_i\vartheta_i(t)>x,\ Y_i\vartheta_i(t)>y \right\}\sim \widetilde{C}_d\bar{F}(x)\bar{G}(y) E\widehat{\vartheta}_{3i}^{\alpha+\beta}(t)$ . 注意到 $(X_i,Y_i,\theta_i)$ 服从 Sarmanov 分布, 即假设 2.1, 于是, $P\left\{ X_i\vartheta_i(t)>x,\ Y_i\vartheta_i(t)>y \right\}$ 可重写为

$P\left\{ X_i\vartheta_i(t)>x,\ Y_i\vartheta_i(t)>y \right\} =E\left\{P\left( X_i\vartheta_i(t)>x,\ Y_i\vartheta_i(t)>y\mid \theta_1,\cdots,\theta_{i-1} \right)\right\}.$

根据引理 4.2, (4.15), (4.19), (4.7), (4.11) 和 (4.12) 式得到

$\begin{matrix} &P\left\{ X_i\vartheta_i(t)>x,\ Y_i\vartheta_i(t)>y \right\} \\ =\ &f_0P\left\{X_i^{\,*}\vartheta_i^{\,*}(t)>x,Y_i^{\,*}\vartheta_i^*(t)>y\right\} -(f_{12}+f_{13})P\left\{\widetilde{X}_i^{\,*}\vartheta_i^*(t)>x,Y_i^{\,*}\vartheta_i^*(t)>y \right\}\nonumber\\ & -(f_{12}+f_{23})P\left\{X^{\,*}_i\vartheta_i^*(t)>x,\widetilde{Y}_i^{\,*}\vartheta_i^*(t)>y\right\} \nonumber\\ & -(f_{13}+f_{23})P\left\{X^{\,*}_i\vartheta_i^*(\widetilde{\theta}_i^*;t)>x,Y_i^*\vartheta_i^*(\widetilde{\theta}_i^*;t)>y\right\} +f_{12}P\left\{\widetilde{X}_i^{\,*}\vartheta_i^*(t)>x,\widetilde{Y}_i^{\,*}\vartheta_i^*(t)>y\right\} \nonumber\\ & +f_{13}P\left\{\widetilde{X}_i^{\,*}\vartheta_i^*(\widetilde{\theta}_i^*;t)>x,Y_i^{\,*}\vartheta_i^*(\widetilde{\theta}_i^*;t)>y\right\} +f_{23}P\left\{X_i^{\,*}\vartheta_i^*(\widetilde{\theta}_i^*;t)>x,\widetilde{Y}_i^{\,*}\vartheta_i^*(\widetilde{\theta}_i^*;t)>y\right\}\nonumber\\ \sim\,& f_0\bar{F}(x)\bar{G}(y)E\vartheta^{*\alpha+\beta}_i(t) -(f_{12}+f_{13})\left(1-d_1/b_1\right)\bar{F}(x)\bar{G}(y)E\vartheta^{*\alpha+\beta}_i(t) \nonumber\\ & -(f_{12}+f_{23})\left(1-d_2/b_2\right)\bar{F}(x)\bar{G}(y)E\vartheta^{*\alpha+\beta}_i(t) -(f_{13}+f_{23})\bar{F}(x)\bar{G}(y) E\left[\vartheta_i^{*\alpha+\beta}(\widetilde{\theta}_i^*;t)\right] \nonumber\\ & +f_{12}\left(1-d_1/b_1\right) \left(1-d_2/b_2\right)\bar{F}(x)\bar{G}(y)E\vartheta^{*\alpha+\beta}_i(t) \nonumber\\ & +f_{13}\left(1-d_1/b_1\right)\bar{F}(x)\bar{G}(y) E\left[\vartheta_i^{*\alpha+\beta}(\widetilde{\theta}_i^*;t)\right] +f_{23}\left(1-d_2/b_2\right)\bar{F}(x)\bar{G}(y) E\left[\vartheta_i^{*\alpha+\beta}(\widetilde{\theta}_i^*;t)\right] \nonumber\\ =\ &\widetilde{C}_d\bar{F}(x)\bar{G}(y) \int_0^t \varphi_{33}(u)E \left[ \mathrm{e}^{-(\alpha + \beta)\xi(\tau_{i - 1}^* +u)}\mathbb{I}_{[\tau_{i - 1}^* +u\leq t]}\right] H(\mathrm{d}u)\nonumber\\ =\ & \widetilde{C}_d \bar{F}(x)\bar{G}(y)E\widehat{\vartheta}_{3i}^{\alpha + \beta}(t), \end{matrix}$

(4.21)

其中 $\vartheta_i^*(\cdot;t)$ 由 (4.17) 式引入.

对 $i<j$ , 证明 $P\left\{ X_i\vartheta_i(t)>x,\ Y_j\vartheta_j(t)>y \right\}$ . 由于 $(X_i,\theta_i)$ 服从二元 Sarmanov 分布, 即 (2.7) 式, 故在引理 4.2 中取 $\eta_{12}=\eta_{23}=0$ 得到

$\begin{matrix} &P\left\{ X_i\vartheta_i(t)>x,\ Y_j\vartheta_j(t)>y \right\} \nonumber\\ =\,&(1+\eta_{13}b_1b_3)P\left\{X_i^{\,*}\vartheta_i^*(t)>x, Y_j\vartheta_j^{\,*}(\theta_j;t)>y\right\}-\eta_{13}b_1b_3P\left\{\widetilde{X}_i^{\,*}\vartheta_i^*(t)>x, Y_j\vartheta_j^{*}(\theta_j;t)>y\right\}\nonumber\\ & -\eta_{13}b_1b_3P\left\{X_i^{\,*}\vartheta_i^*(\widetilde{\theta}_i^*;t)>x, Y_j\vartheta_j^*(\widetilde{\theta}_i^*,\theta_j;t)>y\right\} \nonumber\\ &+\eta_{13}b_1b_3P\left\{\widetilde{X}_i^{\,*}\vartheta_i^*(\widetilde{\theta}_i^*;t)>x, Y_j\vartheta_j^*(\widetilde{\theta}_i^*,\theta_j;t)>y\right\}\nonumber\\ :=\,&(1+\eta_{13}b_1b_3)L_1-\eta_{13}b_1b_3L_2-\eta_{13}b_1b_3L_3+\eta_{13}b_1b_3L_4, \end{matrix}$

(4.22)

其中, $\vartheta_j^*(\cdot;t)$ 和 $\vartheta_j^*(\cdot,\cdot;t)$ 分别由 (4.17) 和 (4.16) 式引入.

考虑 $L_1$ , 由于 $(Y_j,\theta_j)$ 服从二元 Sarmanov 分布, 即 (2.8) 式, 故在引理 4.2 中取 $\eta_{12}=\eta_{13}=0$ 得到

$\begin{matrix} L_1 =\,&(1+\eta_{23}b_2b_3)P\left\{X^{\,*}_i\vartheta_i^*(t)>x,Y_j^{\,*}\vartheta_j^*(t)>y\right\} -\eta_{23}b_2b_3P\left\{X^{\,*}_i\vartheta_i^*(t)>x,\widetilde{Y}_j^{\,*}\vartheta_j^*(t)>y\right\} \nonumber\\ & - \eta_{23}b_2b_3P \left\{ X_i^{\,*}\vartheta_i^*(t) > x,Y_j^{\,*}\vartheta_j^*(\widetilde{\theta}_j^* ; t) > y \right\} + \eta_{23}b_2b_3 P \left\{ X_i^{\,*}\vartheta_i^*(t) > x, \widetilde{Y}_j^{\,*}(\widetilde{\theta}_j^* ; t) > y \right\}, \end{matrix}$

(4.23)

类似于 (4.21) 式的讨论可得 $L_1\sim\bar{F}(x)\bar{G}(y)E\left\{\vartheta^{*\alpha}_{i}(t)\widehat{\vartheta}^{\beta}_{2j}(t)\right\}$ . 由 (4.20) 式及类似于 $L_1$ 中的讨论可得 $L_3\sim\bar{F}(x)\bar{G}(y) E\left\{\vartheta^{*\alpha}_{i}(\widetilde{\theta}^*_i;t) \vartheta^{*\beta}_{j}(\widetilde{\theta}^*_i,\widehat{\theta}_2;t)\right\}$ , 其中, $\widehat{\theta}_2$ 由(2.12)式引入. 类似地, $L_2\sim\left(1- \frac{d_1 }{b_1}\right)\bar{F}(x)\bar{G}(y)E\left\{\vartheta^{*\alpha}_{i}(t)\widehat{\vartheta}^{\beta}_{2j}(t)\right\}$ 和 $L_4\sim\left(1- \frac{d_1 }{b_1}\right)\bar{F}(x)\bar{G}(y) E\left\{\vartheta^{*\alpha}_{i}(\widetilde{\theta}^*_i;t) \vartheta^{*\beta}_{j}(\widetilde{\theta}^*_i,\widehat{\theta}_2;t)\right\}$ . 将 $L_1$ , $L_2$ , $L_3$ 和 $L_4$ 代入 (4.22) 式, 并根据 (2.10) 及 (2.12)-(2.14) 式得到当 $i<j$ 时, $P\{ X_i\vartheta_i(t)>x,$ $Y_j\vartheta_j(t)>y\} \sim \bar{F}(x)\bar{G}(y)E\left\{ \widehat{\vartheta}^{\alpha}_{1i}(t)\widehat{\vartheta}^{\beta}_{4j}(t) \right\}$ . 当 $i>j$ 时, 类似于 $i<j$ 的讨论, 证得 $P\{ X_i\vartheta_i(t)>x,$ $Y_j\vartheta_j(t)>y \} \sim \bar{F}(x)\bar{G}(y) E\left\{\widehat{\vartheta}^{\beta}_{2j}(t)\widehat{\vartheta}^{\alpha}_{5i}(t) \right\}$ . 引理得证.

引理4.4 在定理 2.1 的条件下, 对任意的 $N$ 和 $0<t\leq T$ ,

$\begin{gather*} P\left\{\sum\limits_{i=1}^{N}X_i\vartheta_i(t)>x,\sum\limits_{j=1}^{N}Y_j\vartheta_j(t)>y \right\} \sim \sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}P\left\{X_i\vartheta_i(t)>x,Y_j\vartheta_j(t)>y \right\}. \end{gather*}$

证首先证明概率 $P\Big\{\sum\limits_{i=1}^{N}X_i\vartheta_i(t)>x,\sum\limits_{j=1}^{N}Y_j\vartheta_j(t)>y \Big\}$ 的渐近上界. 对任意的 $\varepsilon>0$ 满足 $\alpha+\beta+2\varepsilon<\kappa$ , 取 $0<\delta<1$ 使得 $(1-\delta)^{-(\alpha+\beta)}<1+\varepsilon$ 成立. 令

$E =\left(\bigcup_{l=1}^{N}\left[X_l\vartheta_l(t)>(1-\delta)x\right]\right)\bigcap\left(\bigcup_{m=1}^{N}\left[ Y_m\vartheta_m(t)>(1-\delta)y\right]\right),$

则

$\begin{matrix} &P\left\{\sum\limits_{i=1}^{N}X_i\vartheta_i(t)>x,\sum\limits_{j=1}^{N}Y_j\vartheta_j(t)>y \right\} =P\left\{\sum\limits_{i=1}^{N}X_i\vartheta_i(t)>x,\sum\limits_{j=1}^{N}Y_j\vartheta_j(t)>y, E\cup E^c \right\} \nonumber\\ \leq\ & \sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}P\left\{X_i\vartheta_i(t)>(1-\delta)x,Y_j\vartheta_j(t)>(1-\delta)y \right\}\nonumber\\ &+\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}\sum\limits_{1\leq m\leq N,\atop{m\neq i}} P\left\{X_i\vartheta_i(t)>x/N, Y_j\vartheta_j(t)>y/N,\, X_m\vartheta_m(t)>\delta x/(N-1) \right\}\nonumber\\ &+\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}\sum\limits_{1\leq m\leq N,\atop{m\neq j}} P\left\{X_i\vartheta_i(t)>x/N,Y_j\vartheta_j(t)>y/N,\,Y_m\vartheta_m(t)>\delta y/(N-1) \right\}\nonumber\\ :=\ &K_1+K_2+K_3. \end{matrix}$

(4.24)

考虑 $K_1$ , 由引理 4.3, $F\in\mathcal{R}_{-\alpha}$ 及 $G\in\mathcal{R}_{-\beta}$ 知对足够大的 $x$ 和 $y$ 有

$\begin{matrix} K_1 &=\left(\sum\limits_{i=j=1}^{N}+\sum\limits_{j=1}^{N-1}\sum\limits_{i=j+1}^{N} +\sum\limits_{i=1}^{N-1}\sum\limits_{j=i+1}^{N}\right) P\left\{X_i\vartheta_i(t)>(1-\delta)x,Y_j\vartheta_j(t)>(1-\delta)y \right\} \nonumber\\ &\sim(1-\delta)^{-(\alpha+\beta)}\bar{F}(x)\bar{G}(y) \left\{\widetilde{C}_d \sum\limits_{i=1}^{N} E\left(\widehat{\vartheta}^{\alpha+\beta}_{3i}(t)\right) +\sum\limits_{j=1}^{N-1}\sum\limits_{i=j+1}^{N} E\left(\widehat{\vartheta}^{\beta}_{2j}(t)\widehat{\vartheta}^{\alpha}_{5i}(t)\right) \right.\nonumber\\ &\quad\ \left.+\sum\limits_{i=1}^{N-1}\sum\limits_{j=i+1}^{N} E\left(\widehat{\vartheta}^{\alpha}_{1i}(t)\widehat{\vartheta}^{\beta}_{4j}(t)\right) \right\}\nonumber\\ & \sim(1-\delta)^{-(\alpha+\beta)}\sum\limits_{i=1}^{N} \sum\limits_{j=1}^{N}P\left\{X_i\vartheta_i(t)>x,Y_j\vartheta_j(t)>y\right\} \nonumber\\ &\leq(1+\varepsilon)\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N} P\left\{X_i\vartheta_i(t)>x,Y_j\vartheta_j(t)>y\right\} \nonumber\\ &\leq\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N} P\left\{X_i\vartheta_i(t)>x,Y_j\vartheta_j(t)>y\right\} +C\varepsilon\bar{F}(x)\bar{G}(y). \end{matrix}$

(4.25)

考虑 $K_2$ , 有

$\begin{align*} K_2=&\left(\sum\limits_{1\leq m <i=j\leq N}+\sum\limits_{1\leq i=j<m\leq N}+\sum\limits_{1\leq i<j=m\leq N} +\sum\limits_{1\leq j=m<i\leq N}+\sum\limits_{1\leq m <i<j\leq N} \right.\\ &\left.+\sum\limits_{1\leq i<j<m\leq N}+\sum\limits_{1\leq m <j<i\leq N} +\sum\limits_{1\leq j<m<i\leq N}+\sum\limits_{1\leq j<i<m\leq N}+\sum\limits_{1\leq i<m<j\leq N}\right) \\ &\cdot P\left\{X_i\vartheta_i(t)>x/N,Y_j\vartheta_j(t)>y/N, X_m\vartheta_m(t)>\delta x/(N-1) \right\}\\ :=\ &K_{20}+K_{21}+K_{22}+\cdots+K_{29}. \end{align*}$

因为 $(X_m,\theta_m)$ 服从二元 Sarmanov 分布 (见 (2.7) 式), 故在引理 4.2 中取 $\eta_{12}=\eta_{23}=0$ 有

$\begin{align*} K_{20} \leq\ & (1+\eta_{13}b_1b_3)\sum\limits_{m=1}^{N-1}\sum\limits_{i=m+1}^{N} P\left\{X_i\vartheta_i^*(\theta_i;t)>x/N, Y_i\vartheta_i^*(\theta_i;t)>y/N,\right.\\ &\left. X_m^{\,*}\vartheta^*_m(t)> \delta x/(N-1)\right\} +\eta_{13}b_1b_3\sum\limits_{m=1}^{N-1}\sum\limits_{i=m+1}^{N} P\left\{ X_i\vartheta_i^*(\widetilde{\theta}_m^*,\theta_i;t)>x/N,\right.\\ &\left. Y_i\vartheta_i^*(\widetilde{\theta}_m^*,\theta_i;t)>y/N, \widetilde{X}_m^{\,*}\vartheta^*_m(\widetilde{\theta}_m^*;t) > \delta x/(N-1)\right\}\\ :=\ &(1+\eta_{13}b_1b_3)K_{201}+\eta_{13}b_1b_3K_{202}, \end{align*}$

其中, $\vartheta_i^*(\cdot,\cdot;\,t)$ 和 $\vartheta^*_m(\cdot;\,t)$ 分别由 (4.16) 和 (4.17) 式引入. 在 $K_{201}$ 中, 注意到 $(X_i,Y_i,\theta_i)$ 服从 Sarmanov 分布, 即假设 2.1, 则由引理 4.2, (4.15), (4.19) 和 (4.8) 式得, 对足够大的 $x$ 和 $y$ 有

$\begin{matrix} K_{201}&\leq\sum\limits_{m=1}^{N-1}\sum\limits_{i=m+1}^{N} \bigg\{ f_0P\left\{ X^{\,*}_i\vartheta^*_i(t)>x/N, Y^{\,*}_i\vartheta_i^*(t)>y/N, X_m^{\,*}\vartheta_m^*(t)>\delta x/(N-1) \right\}\nonumber\\ &\quad+f_{12} P\left\{ \widetilde{X}^{\,*}_i\vartheta_i^*(t)>x/N, \widetilde{Y}^{\,*}_i\vartheta_i^*(t)>y/N, X_m^{\,*}\vartheta_m^*(t)>\delta x/(N-1) \right\}\nonumber\\ &\quad+f_{13} P\left\{\widetilde{X}^{\,*}_i\vartheta_i^*(\widetilde{\theta}^*_i;t)>x/N, Y^{\,*}_i\vartheta_i^*(\widetilde{\theta}_i^*;t)>y/N, X_m^{\,*}\vartheta_m^*(t)>\delta x/(N-1) \right\}\nonumber\\ &\quad+\left.f_{23} P\left\{X^{\,*}_i\vartheta_i^*(\widetilde{\theta}^*_i;t)>x/N, \widetilde{Y}^{\,*}_i\vartheta_i^*(\widetilde{\theta}^*_i;t)>y/N, X_m^{\,*}\vartheta_m^*(t)>\delta x/(N-1) \right\}\right\}\nonumber\\ &\leq C\varepsilon\bar{F}(x)\bar{G}(y)\nonumber. \end{matrix}$

由 (4.11), (4.19) 和 (4.20) 式, 且采用类似于 $K_{201}\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ 中的证明步骤得, 对足够大的 $x$ 和 $y$ 有 $K_{202}\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ . 故对足够大的 $x$ 和 $y$ , $K_{20}\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ . 类似地, 对足够大的 $x$ 和 $y$ , 当 $k=1,2,\cdots,9$ 时, $K_{2k}\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ . 将 $K_{2k}\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ , $k=1,2,\cdots,9$ , 代入 $K_2$ 得 $K_{2} \leq C\varepsilon\bar{F}(x)\bar{G}(y)$ , 对足够大的 $x$ 和 $y$ 成立. 同样地, $K_{3} \leq C\varepsilon\bar{F}(x)\bar{G}(y)$ 对足够大的 $x$ 和 $y$ 成立.

其次证明概率 $P\Big\{\sum\limits_{i=1}^{N}X_i\vartheta_i(t)>x, \sum\limits_{j=1}^{N}Y_j\vartheta_j(t)>y \Big\}$ 的渐近下界. 于是,

$\begin{matrix} &P\bigg\{\sum\limits_{i=1}^{N}X_i\vartheta_i(t)>x,\sum\limits_{j=1}^{N}Y_j\vartheta_j(t)>y \bigg\}\nonumber\\ \geq\ & P\bigg\{\bigcup_{i=1}^{N}\left(X_i\vartheta_i(t)>x\right), \bigcup_{j=1}^{N}\left(Y_j\vartheta_j(t)>y\right) \bigg\}\nonumber\\ \geq\ & \sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N} P\left\{X_i\vartheta_i(t)>x,Y_j\vartheta_j(t)>y \right\} \nonumber\\ & -\sum\limits_{i=1}^N\sum\limits_{j=1}^N\sum\limits_{1\leq m\leq N\atop{i\neq m}} P\left\{X_i\vartheta_i(t)>x,Y_j\vartheta_j(t)>y, X_m\vartheta_m(t)>x \right\}\nonumber\\ & -\sum\limits_{i=1}^N\sum\limits_{j=1}^N\sum\limits_{1\leq m\leq N\atop{j\neq m}} P\left\{X_i\vartheta_i(t)>x,Y_j\vartheta_j(t)>y,Y_m\vartheta_m(t)>y \right\}\nonumber\\ :=\ &{K}{'}_1-{K}'_2-{K}'_3. \end{matrix}$

(4.26)

类似于 $K_2\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ 的证明, 有 ${K}'_{2}\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ 和 ${K}'_{3}\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ 对足够大的 $x$ 和 $y$ 成立. 结合 (4.24) 和 (4.26) 式即可得到结果. 引理 4.4 得证.

对 $0<r<1$ , $0<l\leq \kappa$ , $0<t\leq T$ , 由 Hölder 不等式和 (4.27) 式得

$\begin{matrix} \sum\limits_{n=1}^{\infty}n^q\left(E\vartheta^{\,l}_{n}(t)\right)^{r} \leq\bigg(\sum\limits_{n=1}^{\infty}n^{(q+1)/r}E\vartheta^{l}_{n}(t)\bigg)^{r} \bigg(\sum\limits_{n=1}^{\infty}1/n^{1/(1-r)}\bigg)^{1-r} <\infty, \quad q\geq0. \end{matrix}$

(4.27)

对任意的 $\varepsilon>0$ 且 $\alpha+\beta+2\varepsilon<\kappa$ , 由 Hölder 不等式, Cr 不等式和 (4.27) 式,

$\begin{matrix} &\sum\limits_{i=1}^{\infty}\sum\limits_{j=1}^{\infty}i^{\,q}j^{\,q} E\left\{ \left[\vartheta_i^{\alpha-\varepsilon}(t)+\vartheta_i^{\alpha+\varepsilon}(t)\right] [\vartheta_j^{\beta-\varepsilon}(t)+\vartheta_j^{\beta+\varepsilon}(t)] \right\}\nonumber\\ \leq\ & \sum\limits_{i=1}^{\infty}\sum\limits_{j=1}^{\infty}i^{\,q}j^{\,q} \left\{E\left(\left[\vartheta_i^{\alpha-\varepsilon}(t)+\vartheta_i^{\alpha+\varepsilon}(t)\right] ^{\frac{\alpha+\beta+2\varepsilon}{\alpha+\varepsilon}}\right)\right\}^{\frac{\alpha+\varepsilon}{\alpha+\beta+2\varepsilon}} \\ &\cdot \left\{E\left(\left[\vartheta_j^{\beta-\varepsilon}(t)+ \vartheta_j^{\beta+\varepsilon}(t)\right]^{ \frac{\alpha+\beta+2\varepsilon}{\beta+\varepsilon}}\right)\right\} ^{\frac{\beta+\varepsilon}{\alpha+\beta+2\varepsilon}} \nonumber\\ \leq\ & C\sum\limits_{i=1}^{\infty}\sum\limits_{j=1}^{\infty}i^{\,q}j^{\,q} \left\{\left(E\left[\vartheta_i^{\frac{\alpha-\varepsilon}{\alpha+\varepsilon}(\alpha+\beta+2\varepsilon)}(t)\right]\right) ^{\frac{\alpha+\varepsilon}{\alpha+\beta+2\varepsilon}} +\left(E\left[\vartheta_i^{\alpha+\beta+2\varepsilon}(t)\right]\right)^{ \frac{\alpha+\varepsilon}{\alpha+\beta+2\varepsilon} } \right\}\nonumber\\ &\cdot\left\{\left(E\left[\vartheta_j^{\frac{\beta-\varepsilon}{\beta+\varepsilon}(\alpha+\beta+2\varepsilon)}(t)\right]\right) ^{\frac{\beta+\varepsilon}{\alpha+\beta+2\varepsilon}} +\left(E\left[\vartheta_j^{\alpha+\beta+2\varepsilon}(t)\right]\right)^{ \frac{\beta+\varepsilon}{\alpha+\beta+2\varepsilon} } \right\} \nonumber\\ <\infty,\ q\geq0. \end{matrix}$

(4.28)

类似地,

$\begin{matrix} \sum\limits_{i=1}^{\infty}\sum\limits_{j=1}^{\infty}i^{\,q}j^{\,q} E\left\{ \left[\vartheta_i^{\alpha-\varepsilon}(t)+\vartheta_i^{\alpha+\varepsilon}(t)\right] \vartheta_j^{\beta+\varepsilon}(t)+\vartheta_i^{\alpha+\varepsilon}(t) \left[\vartheta_j^{\beta-\varepsilon}(t)+\vartheta_j^{\beta+\varepsilon}(t)\right] \right\} <\infty. \end{matrix}$

(4.29)

引理4.5 在定理 (2.1) 的条件下, 当 $0<t\leq T$ 时,

$\begin{equation} \lim\limits_{N\rightarrow\infty}\limsup\limits_{(x,y)\rightarrow(\infty,\infty)} \frac{P\bigg\{\sum\limits_{i=N}^{\infty}X_i\vartheta_i(t)>x, \sum\limits_{j=1}^{\infty}Y_j\vartheta_j(t)>y \bigg\}} {\bar{F}(x)\bar{G}(y)}=0, \end{equation}$

(4.30)

和

$\begin{equation} \lim\limits_{N\rightarrow\infty}\limsup\limits_{(x,y)\rightarrow(\infty,\infty)} \frac{P\bigg\{\sum\limits_{i=1}^{\infty}X_i\vartheta_i(t)>x, \sum\limits_{j=N}^{\infty}Y_j\vartheta_j(t)>y \bigg\}} {\bar{F}(x)\bar{G}(y)}=0. \end{equation}$

(4.31)

证首先证明 (4.30) 式. 取 $M$ 满足 $\sum\limits_{i=1}^{\infty}\frac{1}{i^2}<M$ , 于是

$\begin{align*} &P\bigg\{\sum\limits_{i=N}^{\infty}X_i\vartheta_i(t)>x,\sum\limits_{j=1}^{\infty}Y_j\vartheta_j(t)>y\bigg\} \\ \leq\ & \sum\limits_{i=N}^{\infty}\sum\limits_{j=1}^{\infty}P\left\{ X_i\vartheta_i(t)>x/(i^2M), Y_j\vartheta_j(t)>y/(j^2M)\right\}\\ &=\bigg(\sum\limits_{N\leq i<j}^{\infty}+\sum\limits_{i=j=N}^{\infty}+\sum\limits_{i\geq N,i>j}^{\infty}\bigg)P\left\{ X_i\vartheta_i(t)>x/(i^2M), Y_j\vartheta_j(t)>y/(j^2M)\right\}\\ :=\ &L'_1+L'_2+L'_3. \end{align*}$

考虑 $L'_1$ , 根据 (4.22) 和 (4.23) 式提到的方法重写 $L'_1$ . 具体来说, 因为 $(X_i,\theta_i)$ 服从二元 Sarmanov 分布, 即 (2.7) 式, 故在引理 4.2 中取 $\eta_{12}=\eta_{23}=0$ , 此时可将 $L'_1$ 分成四部分 $L'_{1k}$ , $k=1,2,3,4$ . 又根据 $(Y_j,\theta_j)$ 服从二元 Sarmanov 分布, 即 (2.8) 式, 故在引理 4.2 中取 $\eta_{12}=\eta_{13}=0$ , 此时可将 $L'_{11}$ 分成四部分 $L'_{11k}$ , $k=1,2,3,4$ . 接下来根据 (4.2), (4.3), (4.28), (4.29) 和 (4.27) 式得到

$\begin{align*} {L}'_{111} =\ &\sum\limits_{N\leq i<j}^{\infty}P\left\{ X_i^*\vartheta_i^*(t)>x/(i^2M), Y_j^*\vartheta_j^*(t)>y/(j^2M)\right\}\\ =\ &\sum\limits_{N\leq i<j}^{\infty} \left(\int_0^{x/(i^2MD_F)}\int_0^{y/(j^2MD_G)} +\int_0^{x/(i^2MD_F)}\int_{y/(j^2MD_G)}^{\infty} \right.\\&\left.\quad +\int_{x/(i^2MD_F)}^{\infty}\int_0^{y/(j^2MD_G)} +\int_{x/(i^2MD_F)}^{\infty}\int_{y/(j^2MD_G)}^{\infty}\right) \bar{F} \left(x/(i^2Mu)\right) \\ &\cdot \bar{G} \left(y/(j^2Mv)\right) P\left\{ \vartheta_i^*(t)\in \mathrm{d}u, \vartheta_j^*(t)\in \mathrm{d}v \right\}\\ \leq\ & C\bar{F}(x)\bar{G}(y)\sum\limits_{N\leq i<j}^{\infty} i^{2(\alpha+\varepsilon)}j^{2(\beta+\varepsilon)} \int_0^{\infty}\int_0^{\infty} (u^{\alpha-\varepsilon}\vee u^{\alpha+\varepsilon}) (v^{\beta-\varepsilon}\vee v^{\beta+\varepsilon}) \\ & \cdot P\left\{ \vartheta_i^*(t)\in \mathrm{d}u, \vartheta_j^*(t)\in \mathrm{d}v \right\} \\ & +C\bar{F}(x)\sum\limits_{N\leq i<j}^{\infty} i^{2(\alpha+\varepsilon)} \int_0^{x/(i^2MD_F)}\int_{y/(j^2MD_G)}^{\infty} (u^{\alpha-\varepsilon}\vee u^{\alpha+\varepsilon}) \left( vj^2MD_G/y \right)^{\beta+\varepsilon} \\ & \cdot P\left\{ \vartheta_i^*(t)\in \mathrm{d}u, \vartheta_j^*(t)\in \mathrm{d}v \right\} \\ & +C\bar{G}(y)\sum\limits_{N\leq i<j}^{\infty}j^{2(\beta+\varepsilon)} \int_{x/(i^2MD_F)}^{\infty}\int_0^{y/(j^2MD_G)} (v^{\beta-\varepsilon}\vee v^{\beta+\varepsilon}) \left( ui^2MD_F/x \right)^{\alpha+\varepsilon} \\ & \cdot P\left\{ \vartheta_i^*(t)\in \mathrm{d}u, \vartheta_j^*(t)\in \mathrm{d}v \right\} \\ & +C\sum\limits_{N\leq i<j}^{\infty} \int_{x/(i^2MD_F)}^{\infty}\int_{y/(j^2MD_G)}^{\infty} \left( ui^2MD_F/x \right)^{\alpha+\varepsilon} \left( vj^2MD_G/y \right)^{\beta+\varepsilon} \\ & \cdot P\left\{ \vartheta_i^*(t)\in \mathrm{d}u, \vartheta_j^*(t)\in \mathrm{d}v \right\}\\ \leq\ & C\bar{F}(x)\bar{G}(y)\sum\limits_{N\leq i<j}^{\infty} i^{2(\alpha+\varepsilon)}j^{2(\beta+\varepsilon)} E\left\{ \left[\vartheta_i^{*(\alpha+\varepsilon)}(t)\!+\!\vartheta_i^{*(\alpha-\varepsilon)}(t)\right] \left[\vartheta_j^{*(\beta+\varepsilon)}(t)\!+\!\vartheta_j^{*(\beta-\varepsilon)}(t)\right] \right\}\\ &+C\bar{F}(x)y^{-(\beta+\varepsilon)}\sum\limits_{N\leq i<j}^{\infty} i^{2(\alpha+\varepsilon)}j^{2(\beta+\varepsilon)} E\left\{ \left[\vartheta_i^{*(\alpha+\varepsilon)}(t)+ \vartheta_i^{*(\alpha-\varepsilon)}(t)\right] \vartheta_j^{*(\beta+\varepsilon)}(t)\right\}\\ & +Cx^{-(\alpha+\varepsilon)}\bar{G}(y)\sum\limits_{N\leq i<j}^{\infty} i^{2(\alpha+\varepsilon)}j^{2(\beta+\varepsilon)} E\left\{ \left[\vartheta_j^{*(\beta+\varepsilon)}(t)+ \vartheta_j^{*(\beta+\varepsilon)}(t)\right] \vartheta_i^{*(\alpha+\varepsilon)}(t) \right\}\\ &+Cx^{-(\alpha+\varepsilon)}y^{-(\beta+\varepsilon)} \sum\limits_{N\leq i<j}^{\infty}i^{2(\alpha+\varepsilon)}j^{2(\beta+\varepsilon)} E\left\{ \vartheta_i^{*(\alpha+\varepsilon)}(t)\vartheta_j^{*(\beta+\varepsilon)}(t) \right\}, \end{align*}$

于是, $\lim\limits_{N\rightarrow\infty} \limsup\limits_{(x,y)\rightarrow(\infty,\infty)} \frac{{L}'_{111}}{\bar{F}(x)\bar{G}(y)}=0$ . 类似地, 当 $k=2,3,4$ 时, $\lim\limits_{N\rightarrow\infty}\limsup\limits_{(x,y)\rightarrow(\infty,\infty)}$ $\frac{{L}'_{11k}}{\bar{F}(x)\bar{G}(y)}=0$ . 所以, $\lim\limits_{N\rightarrow\infty} \limsup\limits_{(x,y)\rightarrow(\infty,\infty)} \frac{L'_{11}}{\bar{F}(x)\bar{G}(y)}=0$ . 类似地, 当 $k=2,3,4$ 时, $\lim\limits_{N\rightarrow\infty} \limsup\limits_{(x,y)\rightarrow(\infty,\infty)} \frac{L'_{1k}}{\bar{F}(x)\bar{G}(y)}=0$ . 从而, $\lim\limits_{N\rightarrow\infty} \limsup\limits_{(x,y)\rightarrow(\infty,\infty)} \frac{L'_{1}}{\bar{F}(x)\bar{G}(y)}=0.$ 类似于 $L'_{1}$ 的讨论得 $\lim\limits_{N\rightarrow\infty} \limsup\limits_{(x,y)\rightarrow(\infty,\infty)} \frac{L'_{k}}{\bar{F}(x)\bar{G}(y)}=0,$ $k=2,3.$ 故式 (4.30) 得证. 式 (4.31) 类似可证. 引理 4.5 得证.

4.2 定理 (2.1) 的证明

证首先给出证明定理 (2.1) 所需要的若干关系式. 在 (4.27) 和 (4.27) 式中取 $q=0$ , 当 $0<t\leq T$ 时, 由 Hölder 不等式,

$\begin{matrix} &\quad\ \sum\limits_{i=1}^{\infty}E\left(\widehat{\vartheta}^{\alpha+\beta}_{3i}(t)\right) +\sum\limits_{j=1}^{\infty}\sum\limits_{i=1}^{\infty}E\left(\widehat{\vartheta}^{\beta}_{2j}(t)\widehat{\vartheta}^{\alpha}_{5i}(t)\right) +\sum\limits_{i=1}^{\infty}\sum\limits_{j=1}^{\infty} E\left(\widehat{\vartheta}^{\alpha}_{1i}(t)\widehat{\vartheta}^{\beta}_{4j}(t)\right)\nonumber\\ &\leq b_{33}\sum\limits_{i=1}^{\infty}E\left(\vartheta^{*\alpha+\beta}_{i}(t)\right) +b_{32}b_{31}\sum\limits_{j=1}^{\infty}\sum\limits_{i=1}^{\infty} E\left(\vartheta^{*\beta}_{j}(t)\vartheta^{*\alpha}_{i}(t)\right) +b_{31}b_{32}\sum\limits_{i=1}^{\infty}\sum\limits_{j=1}^{\infty} E\left(\vartheta^{*\alpha}_{i}(t)\vartheta^{*\beta}_{j}(t)\right)\nonumber\\ &\leq b_{33}\sum\limits_{i=1}^{\infty}E\left(\vartheta^{*\alpha+\beta}_{i}(t)\right) +2b_{32}b_{31}\sum\limits_{j=1}^{\infty}\left(E\vartheta^{*\alpha+\beta}_{j}(t)\right)^{\frac{\beta}{\alpha+\beta}}\cdot \sum\limits_{i=1}^{\infty}\left(E\vartheta^{*\alpha+\beta}_{i}(t)\right)^{\frac{\alpha}{\alpha+\beta}} \nonumber\\ <\infty, \end{matrix}$

(4.32)

其中, 上式的第一步是根据 (2.10)-(2.12) 式得到. 对任意的 $\varepsilon>0$ 满足 $\alpha+\beta+2\varepsilon<\kappa$ , 由引理 4.5, (4.32) 和 (4.27) 式得, 存在整数 $N>1$ 及足够大的 $x$ 和 $y$ 有

$\begin{matrix} &\qquad P\left\{\sum\limits_{i=N+1}^{\infty}X_i\vartheta_i(t)>x, \sum\limits_{j=1}^{\infty}Y_j\vartheta_j(t)>y \right\} \leq \varepsilon\bar{F}(x)\bar{G}(y), \end{matrix}$

(4.33)

$\begin{matrix} &\qquad P\left\{\sum\limits_{i=1}^{\infty}X_i\vartheta_i(t)>x, \sum\limits_{j=N+1}^{\infty}Y_j\vartheta_j(t)>y \right\} \leq \varepsilon\bar{F}(x)\bar{G}(y), \end{matrix}$

(4.34)

$\begin{matrix} &\sum\limits_{i=N+1}^{\infty}E\widehat{\vartheta}^{\alpha+\beta}_{3i}(t) +\left(\sum\limits_{j=1}^{\infty}\sum\limits_{i=N+1}^{\infty} +\sum\limits_{j=N+1}^{\infty}\sum\limits_{i=j+1}^{\infty}\right) E\left(\widehat{\vartheta}^{\beta}_{2j}(t)\widehat{\vartheta}^{\alpha}_{5i}(t)\right) <\varepsilon, \end{matrix}$

(4.35)

$\begin{matrix} &\left(\sum\limits_{i=1}^{\infty}\sum\limits_{j=N+1}^{\infty} +\sum\limits_{i=N+1}^{\infty}\sum\limits_{j=i+1}^{\infty}\right) E\left(\widehat{\vartheta}^{\alpha}_{1i}(t)\widehat{\vartheta}^{\beta}_{4j}(t)\right) <\varepsilon, \quad \sum\limits_{i=N+1}^{\infty}\frac{1}{i^2}<1. \end{matrix}$

(4.36)

根据定义 2.1, 取 $v_1, v_2\in(1/2,1)$ 使得下式对足够大的 $x$ 和 $y$ 成立

$\begin{matrix} P(X>v_1x)\leq (1+\varepsilon)P(X>x) \;\text{ 和 }\; P(Y>v_2y)\leq (1+\varepsilon)P(Y>y). \end{matrix}$

(4.37)

令 $D_{0,t}^{(i)} =\sup\limits_{0\leq s\leq t}|\int_0^s\mathrm{e}^{-\xi(v)}\mathrm{d}B_i(v)|$ , $i=1,2,3$ . 则根据 Hölder 不等式, Burkholder-Davis-Gundy 不等式, Fubini 定理及定理 (2.1) 中的 $\int_0^TE\mathrm{e}^{-\kappa\xi(s)}\mathrm{d}s<\infty$ 得到

$\begin{matrix} ED_{0,t}^{(i)l} &\leq \left\{ED_{0,t}^{(i)\kappa}\right\}^{l/\kappa} \nonumber\\ \leq C\left\{E\left(\int_0^t\mathrm{e}^{-2\xi(s)}{\rm d}s\right)^{\kappa/2}\right\}^{l/\kappa} \leq C\left\{t^{\kappa/2-1}E\int_0^t\mathrm{e}^{-\kappa\xi(s)}{\rm d}s\right\}^{l/\kappa}\nonumber\\ &=C\left\{t^{\kappa/2-1}\int_0^tE\mathrm{e}^{-\kappa\xi(s)}{\rm d}s\right\}^{l/\kappa} <\infty,\quad 0<l\leq\kappa,\ 0<t\leq T. \end{matrix}$

(4.38)

在 (4.14) 式中取 $l=1$ , 则

$\begin{matrix} E\left(\int_0^t\mathrm{e}^{-\xi(s)}\mathrm{d}s\right) =\int_0^tE\mathrm{e}^{-\xi(s)}\mathrm{d}s<\infty, \quad 0<t\leq T. \end{matrix}$

(4.39)

故由 (4.38), (4.39) 和 (4.15) 式及 Hölder 不等式, 存在足够大的 $A>0$ 使得, 对 $1\leq m,n\leq N$ , $0<l\leq\kappa$ , $0<t\leq T$ 和 $i=1,2,3$ , 有

$\begin{matrix} &P\left\{D_{0,t}^{(i)}>A\right\}<\frac{\varepsilon}{4N^2}, \quad P\left\{\int_0^t\mathrm{e}^{-\xi(s)}{\rm d}s>A\right\}<\frac{\varepsilon}{4N^{\,2}}, \end{matrix}$

(4.40)

$\begin{matrix} E\left\{\vartheta_m^{*l}(t)\left[ \mathbb{I}_{[D_{0,t}^{(i)}>A]} + \mathbb{I}_{[\int_0^t\mathrm{e}^{-\xi(s)}{\rm d}s>A]} \right]\right\} <\frac{\varepsilon}{4N^{\,\kappa+2}}, \end{matrix}$

(4.41)

$\begin{matrix} E\left\{\vartheta_m^{*\alpha+\varepsilon}(t)\vartheta_n^{*\beta+\varepsilon}(t) \left[ \mathbb{I}_{[D_{0,t}^{(i)}>A]} + \mathbb{I}_{[\int_0^t\mathrm{e}^{-\xi(s)}{\rm d}s>A]} \right] \right\}<\frac{\varepsilon}{4N^{\,2\kappa+2}}. \end{matrix}$

(4.42)

由 Hölder 不等式, (4.19) 和 (4.40) 式知, 当 $1\leq k\leq m\leq N$ , $0<l\leq\kappa$ 时, 有

$\begin{matrix} &E\left\{\vartheta_m^{*l}(\widetilde{\theta}^*_k;t)\mathbb{I}_{[D_{0,t}^{(1)}> A]}\right\} \leq \left\{E[\vartheta_m^{*\kappa}(\widetilde{\theta}^*_k;t)]\right\}^{l/\kappa} [P(D_{0,t}^{(1)}> A)]^{(\kappa-l)/\kappa}<\frac{C\varepsilon}{4N^{\,\kappa+2}}. \end{matrix}$

(4.43)

由 Hölder 不等式, (4.15) 和 (4.43) 式得, 当 $1\leq k\leq m\leq N$ 时, 有

$\begin{matrix} E\left\{\vartheta_m^{*\alpha+\varepsilon}(\widetilde{\theta}^*_k;t) \vartheta_n^{*\beta+\varepsilon}(t)\mathbb{I}_{[D_{0,t}^{(1)}> A]}\right\} <\frac{C\varepsilon}{4N^{\,2\kappa+2}}. \end{matrix}$

(4.44)

现在, 分成两步证明有限时间破产概率 $\Psi(x,y;t)$ 的渐近估计式.

第一步证明 $\Psi(x,y;t)$ 的渐近下界. 对满足(4.33)-(4.36) 式的 $N$ , 满足 (4.40)-(4.42) 式的 $A$ 及 $0<t\leq T$ , 有

$\begin{matrix} \Psi(x,y;t) \geq\ & P\left\{ \sum\limits_{i=1}^{\infty}X_i\vartheta_i(t) -c_1\int_0^t\mathrm{e}^{-\xi(s)}\mathrm{d}s -\sigma_1\int_0^t\mathrm{e}^{-\xi(s)}\mathrm{d}B_1(s)>x,\right. \\ &\left. \sum\limits_{j=1}^{\infty}Y_j\vartheta_j(t) c_2\int_0^t\mathrm{e}^{-\xi(s)}\mathrm{d}s- \sigma_2\int_0^t\mathrm{e}^{-\xi(s)}\mathrm{d}B_2(s)>y \right\} \nonumber\\ \geq\ & P\left\{ \sum\limits_{i=1}^{N}X_i\vartheta_i(t)>x+(c_1+\sigma_1)A, \right.\nonumber\\ &\left. \sum\limits_{j=1}^{N}Y_j\vartheta_j(t)>y+(c_2+\sigma_2)A, D_{0,t}^{(1)}\leq A,D_{0,t}^{(2)}\leq A, \int_0^t\mathrm{e}^{-\xi(s)}\mathrm{d}s\leq A\right\}\nonumber\\ \geq\ & P\left\{ \sum\limits_{i=1}^{N}X_i\vartheta_i(t)>x+(c_1+\sigma_1)A,\sum\limits_{j=1}^{N}Y_j\vartheta_j(t)>y+(c_2+\sigma_2)A\right\}\nonumber\\ &-P\left\{ \sum\limits_{i=1}^{N}X_i\vartheta_i(t)>x+(c_1+\sigma_1)A,\sum\limits_{j=1}^{N}Y_j\vartheta_j(t)>y+(c_2+\sigma_2)A, D_{0,t}^{(1)}>A\right\}\nonumber\\ &-P\left\{ \sum\limits_{i=1}^{N}X_i\vartheta_i(t)>x+(c_1+\sigma_1)A,\sum\limits_{j=1}^{N}Y_j\vartheta_j(t)>y+(c_2+\sigma_2)A, D_{0,t}^{(2)}>A\right\}\nonumber\\ &-P\left\{ \sum\limits_{i=1}^{N}X_i\vartheta_i(t)>x+(c_1+\sigma_1)A,\sum\limits_{j=1}^{N}Y_j\vartheta_j(t)>y+(c_2+\sigma_2)A, \right. \\ &\left. \int_0^t\mathrm{e}^{-\xi(s)}\mathrm{d}s>A\right\} \nonumber\\ :={I}'_1-{I}'_2-{I}'_3-{I}'_4. \end{matrix}$

(4.45)

考虑 $I'_1$ , 取 $v_3$ 满足 $v_3x>(c_1+\sigma_1)A$ , $v_3y>(c_2+\sigma_2)A$ 和 $(1+v_3)^{-(\alpha+\beta)}\geq1-\varepsilon$ . 由引理 4.4, 引理 4.3, (4.35) 和 (4.36) 式得对足够大的 $x$ 和 $y$ ,

$\begin{matrix} {I}'_1 \gtrsim\ &\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N} P\left\{X_i\vartheta_i(t)>(1+v_3)x, Y_j\vartheta_j(t)>(1+v_3)y\right\} \sim(1+v_3)^{-(\alpha+\beta)}\bar{F}(x)\bar{G}(y)\nonumber\\ &\cdot\bigg\{ \widetilde{C}_d \sum\limits_{i=1}^{N} E\left(\widehat{\vartheta}^{\alpha+\beta}_{3i}(t)\right) +\sum\limits_{j=1}^{N-1}\sum\limits_{i=j+1}^{N} E\left(\widehat{\vartheta}^{\beta}_{2j}(t)\widehat{\vartheta}^{\alpha}_{5i}(t)\right) +\sum\limits_{i=1}^{N-1}\sum\limits_{j=i+1}^{N} E\left(\widehat{\vartheta}^{\alpha}_{1i}(t)\widehat{\vartheta}^{\beta}_{4j}(t)\right) \bigg\}\nonumber\\ \geq\ &(1-\varepsilon)\bar{F}(x)\bar{G}(y)\bigg\{ \widetilde{C}_d\bigg(\sum\limits_{i=1}^{\infty}-\sum\limits_{i=N+1}^{\infty}\bigg) E\left(\widehat{\vartheta}^{\alpha+\beta}_{3i}(t)\right) \nonumber\\ & +\bigg(\sum\limits_{j=1}^{\infty}\sum\limits_{i=j+1}^{\infty} -\sum\limits_{j=1}^{N}\sum\limits_{i=N+1}^{\infty} \sum\limits_{j=N+1}^{\infty}\sum\limits_{i=j+1}^{\infty}\bigg) E\left(\widehat{\vartheta}^{\beta}_{2j}(t)\widehat{\vartheta}^{\alpha}_{5i}(t)\right) \nonumber\\ & +\bigg(\sum\limits_{i=1}^{\infty}\sum\limits_{j=i+1}^{\infty} -\sum\limits_{i=1}^{N}\sum\limits_{j=N+1}^{\infty} -\sum\limits_{i=N+1}^{\infty}\sum\limits_{j=i+1}^{\infty}\bigg) E\left(\widehat{\vartheta}^{\alpha}_{1i}(t)\widehat{\vartheta}^{\beta}_{4j}(t)\right)\bigg\} \nonumber\\ \geq\ &(1-\varepsilon)\bar{F}(x)\bar{G}(y)\bigg\{\widetilde{C}_d \sum\limits_{i=1}^{\infty} E\left(\widehat{\vartheta}^{\alpha+\beta}_{3i}(t)\right) +\sum\limits_{j=1}^{\infty}\sum\limits_{i=j+1}^{\infty} E\left(\widehat{\vartheta}^{\beta}_{2j}(t)\widehat{\vartheta}^{\alpha}_{5i}(t)\right) \nonumber\\ & +\sum\limits_{i=1}^{\infty}\sum\limits_{j=i+1}^{\infty} E\left(\widehat{\vartheta}^{\alpha}_{1i}(t)\widehat{\vartheta}^{\beta}_{4j}(t)\right)\bigg\} -C\varepsilon\bar{F}(x)\bar{G}(y) \\ :=\ &(1-\varepsilon)\bar{F}(x)\bar{G}(y)\left[I'_{11}+I'_{12}+I'_{13}\right]-C\varepsilon\bar{F}(x)\bar{G}(y). \end{matrix}$

(4.46)

对 $I'_{11}$ 和 $I'_{12}$ , 由 (2.13), (2.14) 和 (2.15) 式得

$\begin{align*} I'_{11}&=\widetilde{C}_d\sum\limits_{i=1}^{\infty}E\left(\mathrm{e}^{-(\alpha+\beta)\xi(\widehat\tau_{i}^{(3)})} \mathbb{I}_{[\widehat\tau_{i}^{(3)}\leq t]}\right) =\widetilde{C}_d\sum\limits_{i=1}^{\infty}\int_0^tE\mathrm{e}^{-(\alpha+\beta)\xi(u)}P(\widehat\tau_{i}^{(3)}\in \mathrm{d}u) =\widetilde{C}_d\int_0^tE\mathrm{e}^{-(\alpha+\beta)\xi(u)}\mathrm{d}\widehat{\lambda}_u^{(3)} \end{align*}$

和

$\begin{align*} I'_{12} &=\sum\limits_{i=2}^{\infty}E\left\{\mathrm{e}^{-\beta\xi(\hat\tau_{1}^{(5)})-\alpha\xi(\hat\tau_{i}^{(5)})} \mathbb{I}_{[\hat\tau_{i}^{(5)}\leq t]} \right\} +\sum\limits_{j=2}^{\infty}\sum\limits_{i=j+1}^{\infty}E\left\{\mathrm{e}^{-\beta\xi(\hat\tau_{j}^{(5)})-\alpha\xi(\widehat\tau_{i}^{(5)})} \mathbb{I}_{[\widehat\tau_{i}^{(5)}\leq t]} \right\}\\ &=\sum\limits_{i=2}^{\infty}\iint\nolimits_{u+v\leq t} E\left\{\mathrm{e}^{-\beta\xi(u)-\alpha\xi(u+v)}\right\} P(\widehat\tau_{1}^{(5)}\in du )P(\widehat\tau_{i}^{(5)}-\widehat\tau_{1}^{(5)}\in dv) \\ &\quad\ +\sum\limits_{j=2}^{\infty}\sum\limits_{i=j+1}^{\infty}\iint\nolimits_{u+v\leq t} E\left\{\mathrm{e}^{-\beta\xi(u)-\alpha\xi(u+v)}\right\} P(\widehat\tau_{j}^{(5)}\in \mathrm{d}u)P(\widehat\tau_{i}^{(5)}-\widehat\tau_{j}^{(5)}\in \mathrm{d}v)\\ &=\iint\nolimits_{u+v\leq t} E\left\{\mathrm{e}^{-\beta\xi(u)-\alpha\xi(u+v)}\right\}\mathrm{d}\widehat{H}_2(u)(\mathrm{d}\widehat{\lambda}_v^{(1)}-\mathrm{d}\lambda_v) \\ &\quad\ +\iint\nolimits_{u+v\leq t} E\left\{\mathrm{e}^{-\beta\xi(u)-\alpha\xi(u+v)}\right\}\mathrm{d}\widehat{\lambda}_u^{(5)}\mathrm{d}\lambda_v, \end{align*}$

其中, $I'_{12}$ 的第一步用到了下式,

$\begin{align*} &\quad\,\sum\limits_{j=2}^{\infty}\sum\limits_{i=j+1}^{\infty} E\left\{\mathrm{e}^{-\beta\xi(\hat\tau_{j}^{\scriptscriptstyle (2)})-\alpha\xi(\hat\tau_{i}^{\scriptscriptstyle (5)})}\mathbb{I}_{[\widehat\tau_{i}^{(5)}\leq t]} \right\}\\ &=\sum\limits_{j=2}^{\infty}\sum\limits_{i=j+1}^{\infty} \idotsint\limits_{s_1+s_2\cdots+s_i\leq t} E\left\{\mathrm{e}^{-\beta\xi(s_1+s_2\cdots+s_j)-\alpha\xi(s_1+s_2\cdots+s_j+\cdots+s_i)} \right\} \\ &\quad\ \cdot \varphi_{32}(s_j)\varphi_{31}(s_i) \mathrm{d}H(s_1)\cdots \mathrm{d}H(s_i)\\ &=\sum\limits_{j=2}^{\infty}\sum\limits_{i=j+1}^{\infty} \idotsint\limits_{s_1+s_2\cdots+s_i\leq t} E\left\{\mathrm{e}^{-\beta\xi(s_1+s_2\cdots+s_j)-\alpha\xi(s_1+s_2\cdots+s_j+\cdots+s_i)} \right\} \\ &\quad\ \cdot \varphi_{32}(s_1)\varphi_{31}(s_2) \mathrm{d}H(s_1)\cdots \mathrm{d}H(s_i)\\ &=\sum\limits_{j=2}^{\infty}\sum\limits_{i=j+1}^{\infty} E\left\{\mathrm{e}^{-\beta\xi(\hat\tau_{j}^{\scriptscriptstyle (5)})-\alpha\xi(\hat\tau_{i}^{\scriptscriptstyle (5) )}}\mathbb{I}_{[\hat\tau_{i}^{(5)}\leq t]} \right\}. \end{align*}$

类似地,

$\begin{align*} I'_{13} =\ &\iint\nolimits_{u+v\leq t} E\left\{\mathrm{e}^{-\alpha\xi(u)-\beta\xi(u+v)}\right)\mathrm{d}\widehat{H}_1(u)(\mathrm{d}\widehat{\lambda}_v^{(2)}-\mathrm{d}\lambda_v) \\ & +\iint\nolimits_{u+v\leq t} E\left\{\mathrm{e}^{-\alpha\xi(u)-\beta\xi(u+v)}\right)\mathrm{d}\widehat{\lambda}_u^{(4)}\mathrm{d}\lambda_v. \end{align*}$

考虑 $I'_2$ , 有

$\begin{align*} {I}'_{2} &\leq \sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N} P\left\{ X_i\vartheta_i(t)>x/N, Y_j\vartheta_j(t)>y/N,D_{0,t}^{(1)}> A\right\}\\ &=\bigg(\sum\limits_{j=1}^{N-1}\sum\limits_{i=j+1}^{N}+\sum\limits_{i=j=1}^{N}+\sum\limits_{i=1}^{N-1}\sum\limits_{j=i+1}^{N}\bigg) P\left\{X_i\vartheta_i(t)>x/N, Y_j\vartheta_j(t)>y/N,D_{0,t}^{(1)}> A\right\}\\ &:=I'_{21}+I'_{22}+I'_{23}. \end{align*}$

在引理 4.2 中取 $\eta_{12}=\eta_{13}=0$ , 得

$\begin{align*} {I}'_{21} &\leq (1+\eta_{23}b_2b_3)\sum\limits_{j=1}^{N-1}\sum\limits_{i=j+1}^{N} P\left\{X_i\vartheta_i^*(\theta_i;t)I_{[D_{0,t}^{(1)}> A]}>x/N, Y_j^{\,*}\vartheta_j^*(t)\mathbb{I}_{[D_{0,t}^{(1)}> A]}>y/N\right\}\\ &\quad +\eta_{23}b_2b_3\sum\limits_{j=1}^{N-1}\sum\limits_{i=j+1}^{N} P\left\{X_i\vartheta_i^*(\widetilde{\theta}^*_j,\theta_i;t)\mathbb{I}_{[D_{0,t}^{(1)}> A]}>x/N, \widetilde{Y}_j^{\,*}\vartheta_j^*(\widetilde{\theta}^*_j;t)\mathbb{I}_{[D_{0,t}^{(1)}> A]}>y/N\right\}\\ &:=(1+\eta_{23}b_2b_3){I}'_{211}+\eta_{23}b_2b_3{I}'_{212}. \end{align*}$

进一步地, 在引理 4.2 中取 $\eta_{12}=\eta_{23}=0$ , 于是 $I'_{211}$ 可重写为

$\begin{align*} I'_{211} &\leq (1+\eta_{13}b_1b_3)\sum\limits_{j=1}^{N-1}\sum\limits_{i=j+1}^{N} P\left\{X_i^{\,*}\vartheta_i^*(t)\mathbb{I}_{[D_{0,t}^{(1)}> A]}>x/N, Y_j^{\,*}\vartheta_j^*(t)\mathbb{I}_{[D_{0,t}^{(1)}> A]}>y/N\right\}\\ &\quad +\eta_{13}b_1b_3\sum\limits_{j=1}^{N-1}\sum\limits_{i=j+1}^{N} P\left\{\widetilde{X}_i^{\,*}\vartheta_i^*(\widetilde{\theta}^*_i;t)\mathbb{I}_{[D_{0,t}^{(1)}> A]}>x/N, Y_j^*\vartheta_j^{\,*}(t) \mathbb{I}_{[D_{0,t}^{(1)}> A]}>y/N\right\}\\ & :=(1+\eta_{13}b_1b_3){I}'_{2111}+\eta_{13}b_1b_3{I}'_{2112}. \end{align*}$

根据 (4.5), (4.40), (4.41) 和 (4.42) 式得到

$\begin{align*} {I}'_{2111} &=\sum\limits_{j=1}^{N-1}\sum\limits_{i=j+1}^{N} E\Big\{ P\Big\{X_i^{\,*}\vartheta_i^*(t)\mathbb{I}_{[D_{0,t}^{(1)}> A]}>x/N,Y_j^{\,*}\vartheta_j^*(t)\mathbb{I}_{[D_{0,t}^{(1)}> A]}>y/N \mid \vartheta_j^*(t),\\ &\quad\ \vartheta_i^*(t),D_{0,t}^{(1)}\Big\}\Big\}\\ &=\sum\limits_{j=1}^{N-1}\sum\limits_{i=j+1}^{N} E\left\{ P\left\{X_i^{\,*}\vartheta_i^*(t)\mathbb{I}_{[D_{0,t}^{(1)}> A]}>x/N\mid \vartheta_j^*(t),\vartheta_i^*(t),D_{0,t}^{(1)}\right\} \right.\\ &\quad\left. \cdot P\left\{Y_j^{\,*}\vartheta_j^*(t)\mathbb{I}_{[D_{0,t}^{(1)}> A]}>y/N\mid \vartheta_j^*(t),\vartheta_i^*(t),D_{0,t}^{(1)}\right\}\right\}\\ &\leq C\bar{F}(x)\bar{G}(y)\sum\limits_{j=1}^{N-1}\sum\limits_{i=j+1}^{N} \left\{N^{2\kappa}E\left(\vartheta_i^{*\alpha+\varepsilon}(t)\vartheta_j^{*\beta+\varepsilon}(t)\mathbb{I}_{[D_{0,t}^{(1)}> A]}\right) \right.\\ &\left.\quad +N^{\,\kappa}E\left(\vartheta_i^{*\alpha+\varepsilon}(t)\mathbb{I}_{[D_{0,t}^{(1)}> A]}\right) +N^{\,\kappa}E\left(\vartheta_j^{*\beta+\varepsilon}(t)\mathbb{I}_{[D_{0,t}^{(1)}> A]}\right) +P\left(D_{0,t}^{(1)}> A\right) \right\}\\ &\leq C\varepsilon\bar{F}(x)\bar{G}(y), \quad\text{对足够大的} \; x \; \text{和} \;y. \end{align*}$

类似于 ${I}'_{2111}\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ , 由 (4.11), (4.43) 和 (4.44) 式知 ${I}'_{2112}\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ 对足够大的 $x$ 和 $y$ 成立. 故 ${I}'_{211}\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ 对足够大的 $x$ 和 $y$ 成立. 由 Hölder 不等式, (4.20), (4.19), (4.40) 和 (4.43) 式得, 当 $0<l\leq\kappa$ 及 $1\leq n<m\leq N$ 时, $E\big\{\vartheta_m^{*l}(\widetilde{\theta}^*_n,\widetilde{\theta}^*_m;t) \mathbb{I}_{[D_{0,t}^{(1)}> A]}\big\} <\frac{C\varepsilon}{4N^{\,\kappa+2}}$ 且 $E\big\{[\vartheta_m^{*\alpha+\varepsilon}(\widetilde{\theta}^*_m;t)+\vartheta_m^{*\alpha+\varepsilon}(\widetilde{\theta}^*_n, \widetilde{\theta}^*_m;t)]\vartheta_n^{*\beta+\varepsilon}(\widetilde{\theta}^*_n;t)\mathbb{I}_{[D_{0,t}^{(1)}> A]}\big\} <\frac{C\varepsilon}{4N^{\,2\kappa+2}},$ 于是类似于 ${I}'_{211}\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ 的步骤可以得到 ${I}'_{212}\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ 对足够大的 $x$ 和 $y$ 成立, 故 ${I}'_{21}\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ 对足够大的 $x$ 和 $y$ 成立. 类似于 $I'_{21}$ , 当 $k=2,3$ 时, ${I}'_{2k}\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ 对足够大的 $x$ 和 $y$ 成立. 于是,

${I}'_{2}\leq C\varepsilon\bar{F}(x)\bar{G}(y), \quad\text{对足够大的} \; x \; \text{和} \;y.$

类似于 ${I}'_{2}$ ,

${I}'_{3}\leq C\varepsilon\bar{F}(x)\bar{G}(y),\quad {I}'_{4}\leq C\varepsilon\bar{F}(x)\bar{G}(y), \quad \text{对足够大的} \; x \; \text{和} \;y.$

将 $I'_k$ , $k=1,2,3,4$ , 代入 (4.45) 式得到

$\begin{matrix} \Psi(x,y;t) \geq\ & \bar{F}(x)\bar{G}(y) \left\{ \widetilde{C}_d \int_{0}^{t}E\mathrm{e}^{-(\alpha+\beta)\xi(u)} \mathrm{d}\widehat{\lambda}_{u}^{(3)} \right.\nonumber\\ &\left. +\iint_{u+v\leq t} E\mathrm{e}^{-\beta\xi(u)-\alpha\xi(u+v)} [\mathrm{d}\widehat{\lambda}_{u}^{(5)}\mathrm{d}\lambda_v +\mathrm{d}\widehat{H}_2(u) (\mathrm{d}\widehat{\lambda}_{v}^{(1)}-\mathrm{d}\lambda_v)] \right.\nonumber\\ &\left. +\iint_{u+v\leq t} E\mathrm{e}^{-\alpha\xi(u)-\beta\xi(u+v)} [\mathrm{d}\widehat{\lambda}_{u}^{(4)}\mathrm{d}\lambda_v +\mathrm{d}\widehat{H}_1(u) (\mathrm{d}\widehat{\lambda}_{v}^{(2)} -\mathrm{d}\lambda_v)]\right\} \\ & -C\varepsilon\bar{F}(x)\bar{G}(y). \end{matrix}$

(4.47)

第二步证明概率 $\Psi(x,y;t)$ 的渐近上界. 对满足(4.33)-(4.36) 式的 $N$ 和满足 (4.37) 式的 $v_1$ 和 $v_2$ , 有

$\begin{matrix} \Psi(x,y;t)&=P\bigg\{\bigcup_{0<s\leq t}\bigg( \sum\limits_{i=1}^{\infty}X_i\vartheta_i(s) -c_1\int_0^s\mathrm{e}^{-\xi(u)}\mathrm{d}u -\sigma_1\int_0^s\mathrm{e}^{-\xi(u)}\mathrm{d}B_1(u)>x, \nonumber\\ &\quad\ \sum\limits_{j=1}^{\infty}Y_j\vartheta_j(s) -c_2\int_0^s\mathrm{e}^{-\xi(v)}\mathrm{d}v -\sigma_2\int_0^s\mathrm{e}^{-\xi(v)}\mathrm{d}B_2(v)>y \bigg)\bigg\}\nonumber\\ &\leq P\bigg\{ \sum\limits_{i=1}^{\infty}X_i\vartheta_i(t)+\sigma_1D_{0,t}^{(1)}>x, \sum\limits_{j=1}^{\infty}Y_j\vartheta_j(t)+\sigma_2D_{0,t}^{(2)}>y \bigg\}\nonumber\\ &\leq P\bigg\{\sum\limits_{i=1}^{N}X_i\vartheta_i(t)>v_1x, \sum\limits_{j=1}^{N}Y_j\vartheta_j(t)>v_2y\bigg\} \nonumber\\&\quad\ +P\bigg\{ \sum\limits_{i=1}^{N}X_i\vartheta_i(t)>v_1x, \sum\limits_{j=N+1}^{\infty}Y_j\vartheta_j(t)>\frac{1-v_2}{2}y \bigg\}\nonumber\\ & \quad\ +P\bigg\{ \sum\limits_{i=1}^{N}X_i\vartheta_i(t)>v_1x, \sigma_2D_{0,t}^{(2)}>\frac{1-v_2}{2}y\bigg\}\nonumber\\&\quad\ +P\bigg\{ \sum\limits_{i=N+1}^{\infty}X_i\vartheta_i(t)>\frac{1-v_1}{2}x, \sum\limits_{j=1}^{N}Y_j\vartheta_j(t)>v_2y \bigg\}\nonumber\\ & \quad\ +P\bigg\{ \sum\limits_{i=N+1}^{\infty}X_i\vartheta_i(t)>\frac{1-v_1}{2}x, \sum\limits_{j=N+1}^{\infty}Y_j\vartheta_j(t)>\frac{1-v_2}{2}y \bigg\}\nonumber\\& \quad\ +P\bigg\{ \sum\limits_{i=N+1}^{\infty}X_i\vartheta_i(t)>\frac{1-v_1}{2}x, \sigma_2D_{0,t}^{(2)}>\frac{1-v_2}{2}y\bigg\}\nonumber\\&\quad\ +P\bigg\{ \sigma_1D_{0,t}^{(1)}>\frac{1-v_1}{2}x, \sum\limits_{j=1}^{N}Y_j\vartheta_j(t)>v_{2}y\bigg\} \nonumber\\& \quad\ +P\bigg\{ \sigma_1D_{0,t}^{(1)}>\frac{1-v_1}{2}x, \sum\limits_{j=N+1}^{\infty}Y_j\vartheta_j(t)>\frac{1-v_2}{2}y \bigg\}\nonumber\\&\quad\ +P\bigg\{ \sigma_1D_{0,t}^{(1)}>\frac{1-v_1}{2}x, \sigma_2D_{0,t}^{(2)}>\frac{1-v_2}{2}y\bigg\}\nonumber\\ &:=I_1+I_2+I_3+I_4+I_5+I_6+I_7+I_8+I_9. \end{matrix}$

(4.48)

考虑 $I_1$ , 由引理 4.4, 引理 4.3 及 (4.37) 式得

$\begin{align*} I_1&\sim\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}P\left\{X_i\vartheta_i(t)>v_1x,Y_j\vartheta_j(t)>v_2y\right\} \\ & \sim\bar{F}(v_1x)\bar{G}(v_2y) \bigg\{\widetilde{C}_d \sum\limits_{i=1}^{N} E\left(\widehat{\vartheta}^{\alpha+\beta}_{3i}(t)\right) +\sum\limits_{j=1}^{N-1}\sum\limits_{i=j+1}^{N} E\left(\widehat{\vartheta}^{\beta}_{2j}(t)\widehat{\vartheta}^{\alpha}_{5i}(t)\right) \\ &\quad\ +\sum\limits_{i=1}^{N-1}\sum\limits_{j=i+1}^{N} E\left(\widehat{\vartheta}^{\alpha}_{1i}(t)\widehat{\vartheta}^{\beta}_{4j}(t)\right) \bigg\}\\ &\leq (1+\varepsilon)^2\bar{F}(x)\bar{G}(y)\left(I'_{11}+I'_{12}+I'_{13}\right), \quad \text{对足够大的} \; x \; \text{和} \;y, \end{align*}$

其中, $I'_{1k}$ , $k=1,2,3$ , 由 (4.46) 式引入. 考虑 $I_3$ , 在引理 4.2 中取 $\eta_{12}=\eta_{23}=0$ , 则

$\begin{align*} I_3 &\leq (1+\eta_{13}b_1b_3)\sum\limits_{i=1}^N P\left(X_i^{\,*}\vartheta_i^*(t)\mathbb{I}_{[\sigma_2D_{0,t}^{(2)}>(1-v_2)y/2]}>v_1x/N\right) \\ &\quad\ + \eta_{13}b_1b_3\sum\limits_{i=1}^N P\left( \widetilde{X}_i^{\,*}\vartheta_i^*(\widetilde{\theta}_i^*;t) \mathbb{I}_{[\sigma_2D_{0,t}^{(2)}>(1-v_2)y/2]}>v_1x/N\right)\\ &:=(1+\eta_{13}b_1b_3)I_{31}+\eta_{13}b_1b_3I_{32}. \end{align*}$

由 (4.5), Hölder 式不等式, Markov 不等式, (4.38), (4.27) 和 (4.3) 式得

$\begin{align*} I_{31}&=\sum\limits_{i=1}^N E\left\{P\left(X_i^{\,*}\vartheta_i^*(t)\mathbb{I}_{[\sigma_2D_{0,t}^{(2)}>(1-v_2)y/2]}>v_1x/N|\vartheta_i^*(t),D_{0,t}^{(2)}\right)\right\}\\ &\leq C\bar{F}(x)\sum\limits_{i=1}^N\left[N^{\alpha+\varepsilon}E\left(\vartheta_i^{*\alpha+\varepsilon}(t)\mathbb{I}_{[\sigma_2D_{0,t}^{(2)}>(1-v_2)y/2]}\right) +E\mathbb{I}_{[\sigma_2D_{0,t}^{(2)}>(1-v_2)y/2]}\right]\\ &\leq C\bar{F}(x)\left\{N^{\alpha+\varepsilon}\sum\limits_{i=1}^N \left(E\vartheta_i^{*\alpha+\beta+2\varepsilon}(t)\right) ^{\frac{\alpha+\varepsilon}{\alpha+\beta+2\varepsilon}} \left(E\mathbb{I}_{[\sigma_2D_{0,t}^{(2)}>(1-v_2)y/2]}\right) ^{\frac{\beta+\varepsilon}{\alpha+\beta+2\varepsilon}} \right.\\ &\quad\ \left. +\sum\limits_{i=1}^NP\left( \sigma_2D_{0,t}^{(2)}>(1-v_2)y/2 \right)\right\}\\ &\leq C\bar{F}(x)y^{-(\beta+\varepsilon)} \bigg[N^{\alpha+\varepsilon}\left(ED_{0,t}^{(2)\alpha+\beta+2\varepsilon}\right) ^{\frac{\beta+\varepsilon}{\alpha+\beta+2\varepsilon}} \sum\limits_{i=1}^N \left(E\vartheta_i^{*\alpha+\beta+2\varepsilon}(t)\right) ^{\frac{\alpha+\varepsilon}{\alpha+\beta+2\varepsilon}} \\ &\quad\, +\sum\limits_{i=1}^N ED_{0,t}^{(2)\beta+\varepsilon} \bigg]\\ &\leq C\bar{F}(x)y^{-(\beta+\varepsilon)} \leq C\varepsilon\bar{F}(x)\bar{G}(y), \quad\text{对足够大的} \; x \; \text{和} \;y. \end{align*}$

类似地, $I_{32}\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ 对足够大的 $x$ 和 $y$ 成立. 故

$\begin{align*} I_{3}\leq C\varepsilon\bar{F}(x)\bar{G}(y), \quad\text{对足够大的} \; x \; \text{和} \;y. \end{align*}$

类似于 $I_3\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ 的推导过程, 要证明 $I_{6}\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ 对足够大的 $x$ 和 $y$ 成立, 只需证明 $\sum\limits_{i=N+1}^{\infty}P\left( X_i^{\,*}\vartheta_i^*(t) \mathbb{I}_{[\sigma_2D_{0,t}^{(2)}>(1-v_2)y/2]} >\frac{(1-v_1)x}{2i^2}\right) \leq C\varepsilon\bar{F}(x)\bar{G}(y)$ . 而由 (4.5) 式, Hölder 不等式, Markov 不等式, (4.38), (4.27) 和 (4.3) 式得

$\begin{align*} &\quad\ \sum\limits_{i=N+1}^{\infty}E\left\{ P\left(X_i^*\vartheta_i^*(t) \mathbb{I}_{[\sigma_2D_{0,t}^{(2)}>(1-v_2)y/2]} >\frac{(1-v_1)x}{2i^2} \Big|\vartheta_i^*(t),D_{0,t}^{(2)}\right)\right\}\\ &\leq C\bar{F}(x)\bigg[ \sum\limits_{i=N+1}^{\infty}i^{2(\alpha+\varepsilon)}E\left( \vartheta_i^{*\alpha+\varepsilon}(t) \mathbb{I}_{[\sigma_2D_{0,t}^{(2)}>(1-v_2)y/2]}\right)\\ &\quad\ +\sum\limits_{i=N+1}^{\infty}E\left( \mathbb{I}_{[\vartheta_i^*(t)<1]}\mathbb{I}_{[\sigma_2D_{0,t}^{(2)}>(1-v_2)y/2]}\right) \bigg] \\ &\leq C\bar{F}(x)\bigg\{ \sum\limits_{i=N+1}^{\infty}i^{2(\alpha+\varepsilon)} \left(E\vartheta_i^{*\alpha+\beta+2\varepsilon}(t)\right) ^{\frac{\alpha+\varepsilon}{\alpha+\beta+2\varepsilon}} \left[P(\sigma_2D_{0,t}^{(2)}>\frac{(1-v_2)y}{2})\right] ^{\frac{\beta+\varepsilon}{\alpha+\beta+2\varepsilon}}\\ &\quad\ \cdot \sum\limits_{i=N+1}^{\infty} [P(\vartheta_i^*(t)<1)] ^{\frac{\alpha+\varepsilon}{\alpha+\beta+2\varepsilon}} \left[P(\sigma_2D_{0,t}^{(2)}>\frac{(1-v_2)y}{2})\right] ^{\frac{\beta+\varepsilon}{\alpha+\beta+2\varepsilon}} \bigg\}\\ &\leq C\bar{F}(x)y^{-(\beta+\varepsilon)}\bigg\{\sum\limits_{i=N+1}^{\infty}i^{2(\alpha+\varepsilon)} \left(E\vartheta_i^{*\alpha+\beta+2\varepsilon}(t)\right) ^{\frac{\alpha+\varepsilon}{\alpha+\beta+2\varepsilon}} +\sum\limits_{i=N+1}^{\infty}\left[P(\vartheta_i^*(t)<1)\right] ^{\frac{\alpha+\varepsilon}{\alpha+\beta+2\varepsilon}} \bigg\}\\ &\leq C\varepsilon\bar{F}(x)\bar{G}(y), \quad\text{对足够大的} \; x \; \text{和} \;y, \end{align*}$

其中, 上式的最后一步由下式得到: 当 $q\geq0$ , $0<r\leq1$ , $0<t\leq T$ , 由伽马分布和 (4.13) 式,

$\begin{align*} \sum\limits_{i=1}^{\infty}i^q[P(\vartheta_i^*(t)<1)]^{r} &\leq\bigg(\sum\limits_{i=1}^{\infty}i^{(q+1)/r}P(\vartheta_i^*(t)<1)\bigg)^r \bigg(\sum\limits_{i=1}^{\infty}i^{-1/(1-r)}\bigg)^{1-r}\\ &\leq C\bigg[\sum\limits_{i=1}^{\infty}i^{(q+1)/r}\int_0^tP(\mathrm{e}^{-\xi(s)}<1) P(\tau_i\in \mathrm{d}s)\bigg]^{r} \\ &\leq C\left(t^{[(q+1)/r]+1}\vee1\right)^{r}\left(\int_0^tP(\mathrm{e}^{-\xi(s)}<1)\mathrm{d}s\right)^{r} \leq Ct^{\,r}<\infty. \end{align*}$

类似于 $I_3\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ 和 $I_6\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ , 分别得到 $I_7\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ 和 $I_8\leq C\varepsilon\bar{F}(x)\bar{G}(y)$ 对足够大的 $x$ 和 $y$ 成立. 由 (4.5) 和 (4.33) 式, 及定义 2.1 得, 对足够大的 $x$ 和 $y$ , 有 $I_k\leq C\varepsilon\bar{F}(x)\bar{G}(y), \ k=2,4,5,$ 考虑 $I_9$ , 若扰动系数 $\sigma_1=0$ 或 $\sigma_2=0$ , 易得 $I_9\leq C\varepsilon \bar{F}(x)\bar{G}(y)$ 对足够大的 $x$ 和 $y$ 成立, 故考虑 $\sigma_1\neq0$ 且 $\sigma_2\neq0$ . 由于 $B_2(t)=\rho B_1(t)+\sqrt{1-\rho^2}B_3(t)$ , $I_9$ 可改写为

$\begin{align*} I_9 &=P\bigg\{ \sigma_1D_{0,t}^{(1)}>(1-v_1)x/2,\\ &\quad\ \sigma_2\sup\limits_{0\leq s\leq t}\Big| \rho\int_0^s\mathrm{e}^{-\xi(v)}\mathrm{d}B_1(v) +\sqrt{1-\rho^2}\int_0^s\mathrm{e}^{-\xi(v)}\mathrm{d}B_3(v)\Big| > (1-v_2)y/2\bigg\}\\ & \leq P\left\{ D_{0,t}^{(1)}>(1-v_1)x/(2\sigma_1), |\rho| D_{0,t}^{(1)}>(1-v_2)y/(4\sigma_2)\right\} \\ &\quad\ +P\left\{D_{0,t}^{(1)}>(1-v_1)x/(2\sigma_1), \sqrt{1-\rho^2}D_{0,t}^{(3)}>(1-v_2)y/(4\sigma_2)\right\} :=I_{91}+I_{92}. \end{align*}$

若 $\rho\in(-1,0)\cup(0,1)$ , 则由 Markov 不等式, (4.38) 和 (4.3) 式有

$\begin{align*} I_{91} &=P\left\{D_{0,t}^{(1)}>[(1-v_1)x/(2\sigma_1)]\vee [(1-v_2)y/(4\sigma_2|\rho|)]\right\} \\ &\leq C E\left(D_{0,t}^{(1)\alpha+\beta+2\varepsilon}\right)(x^{\alpha+\beta+2\varepsilon}\vee y^{\alpha+\beta+2\varepsilon})^{-1}\\ &\leq C E\left(D_{0,t}^{(1)\alpha+\beta+2\varepsilon}\right)(x^{\alpha+\varepsilon}y^{\beta+\varepsilon})^{-1} \leq C\varepsilon \bar{F}(x)\bar{G}(y) \end{align*}$

和

$\begin{align*} I_{92} &\leq Cx^{-(\alpha+\varepsilon)}y^{-(\beta+\varepsilon)}E\left(D_{0,t}^{(1)\alpha+\varepsilon}D_{0,t}^{(3)\beta+\varepsilon}\right) \\ &\leq Cx^{-(\alpha+\varepsilon)}y^{-(\beta+\varepsilon)} \left\{E\left(D_{0,t}^{(1)\alpha+\beta+2\varepsilon}\right)\right\} ^{\frac{\alpha+\varepsilon}{\alpha+\beta+2\varepsilon}} \left\{E\left(D_{0,t}^{(3)\alpha+\beta+2\varepsilon}\right)\right\} ^{\frac{\beta+\varepsilon}{\alpha+\beta+2\varepsilon}} \leq C\varepsilon \bar{F}(x)\bar{G}(y) \end{align*}$

对足够大的 $x$ 和 $y$ 成立, 其中, $I_{92}$ 的第二步由 Hölder 不等式得到. 若 $\rho=0$ 或 $\pm1$ , 则易得

$I_9\leq C\varepsilon \bar{F}(x)\bar{G}(y)$ 对足够大的 $x$ 和 $y$ 成立. 故

$\begin{align*} I_9\leq C\varepsilon \bar{F}(x)\bar{G}(y), \quad\text{对足够大的} \; x \; \text{和} \;y. \end{align*}$

将 $I_k$ , $k=1,\cdots,9$ , 代入 (4.48) 式得

$\begin{matrix} \Psi(x,y;t) \leq\ & \bar{F}(x)\bar{G}(y) \left\{ \widetilde{C}_d \int_{0}^{t}E\mathrm{e}^{-(\alpha+\beta)\xi(u)} \mathrm{d}\widehat{\lambda}_{u}^{(3)} \right.\nonumber\\ &\left. +\iint_{u+v\leq t} E\mathrm{e}^{-\beta\xi(u)-\alpha\xi(u+v)} [\mathrm{d}\widehat{\lambda}_{u}^{(5)}\mathrm{d}\lambda_v +\mathrm{d}\widehat{H}_2(u) (\mathrm{d}\widehat{\lambda}_{v}^{(1)} -\mathrm{d}\lambda_v)] \right.\nonumber\\ &\left. +\iint_{u+v\leq t} E\mathrm{e}^{-\alpha\xi(u)-\beta\xi(u+v)} [\mathrm{d}\widehat{\lambda}_{u}^{(4)}\mathrm{d}\lambda_v +\mathrm{d}\widehat{H}_1(u)(\mathrm{d}\widehat{\lambda}_{v}^{(2)} -\mathrm{d}\lambda_v)]\right\} \\ &+C\varepsilon\bar{F}(x)\bar{G}(y). \end{matrix}$

(4.49)

结合 (4.47) 和 (4.49) 式即可得到结果, 定理 (2.1) 得证.

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2010

... 近年来, 一维风险模型破产概率的研究成果丰硕^{[1,8,10,11,19,21,23]}. 对于保险公司来说, 只经营一种业务是不大可能的, 因此多维保险风险模型的研究被提上日程, 而二维保险风险模型作为一维保险风险模型的自然推广及高维保险风险模型的特例具有重要意义, 见文献 [3,4,5,6,7,9,12,16,17,18,20,25,27]. 随着我国财富日益增多和金融活动日益频繁, 保险公司资产的投资收益对其风险具有越来越重要的影响, 故在度量保险公司的破产概率时须考虑到资产的投资收益这一重要因素. 很多学者假设保险公司资产的投资收益是常数, 但即使保险公司在债券市场投资, 其收益率也是变化的, 这一理想化假设只是为了方便在数学上处理. 在金融活动日益频繁的今天, 保险公司通常会将投资组合的一部分资产购买可能获得较高回报的风险资产, 因此, 保险公司资产的投资收益具有不确定性与风险性. 也有学者假设保险公司资产的对数投资收益过程由 Lévy 过程刻画, 但对保险公司来说, 其投资收益过程并非全是由平稳独立过程刻画, 例如, CIR 利率模型刻画的投资收益过程是 Markov 过程, 基于分形布朗运动的随机波动性模型刻画的投资收益过程甚至不能由 Markov 过程刻画. 为保证保险公司破产概率的估计不会过低或过高, 如何选择随机过程描述保险公司资产的投资收益具有挑战性, 为解决这一问题, 假设保险公司的对数投资收益过程由一般的右连左极随机过程刻画, 本文研究二维带扰动保险风险模型破产概率的渐近估计. ...

1987

... 根据专著 ^[2], 若

$F\in\mathcal{R}_{-\alpha}$

$\alpha>0$

, 则存在常数

$C_F>1$

和

$D_F>0$

, 使得对任意的

$0<\delta_0<\alpha$

和

$x, xy\geq D_F$

有 ...

Some results on ruin probabilities in a two-dimensional risk model

2003

Asymptotic results for ruin probability of a two-dimensional renewal risk model

2013

Uniform asymptotics for the ruin probabilities in a bidimensional renewal risk model with strongly subexponential claims

2019

Uniform asymptotics for the ruin probabilities of a two-dimensional renewal risk model with dependent claims and risky investments

2017

On a two-dimensional risk model with time-dependent claim sizes and risky investments

2018

Ruin probabilities in the presence of regularly varying tails and optimal investment

2002

Asymptotic ruin probabilities in a generalized bidimensional risk model perturbed by diffusion with constant force of interest

2014

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2013

Tail asymptotic for discounted aggregate claims with one-sided linear dependence and general investment return

2019

Asymptotic results for ruin probability in a two-dimensional risk model with stochastic investment returns

2017

Finite-and infinite-time ruin probabilities with general stochastic investment return processes and bivariate upper tail independent and heavy-tailed claims

2013

... 为得到

$E\{\mathrm{e}^{-C_2(u,u+v)r_u-(a+b)\xi(u)}\}$

, 设

$D'(p_1,p_2,t)=E\mathrm{e}^{p_1r_t+p_2\xi(t)}$

, 其中

$p_1$

和

$p_2$

是实数. 由文献 [13],

$r_t$

和

$\xi(t)=\int_0^tr_s\mathrm{d}s$

皆有矩母函数, 类似于上面与

$D(p_1,p_2,t)$

有关的讨论知, 当

$m^2>2\delta^2p_2$

时, ...

A closed-form solution for options with stochastic volatility with applications to bond and currency options

1993

... 1993 年, Steven Heston 建立了一类随机波动率模型, Heston 模型^[14]. 与 Black-Scholes 模型相比, Heston 模型中波动率是任意的. 令投资收益过程

$\left\{S_t=\mathrm{e}^{\xi(t)},\ t\geq0 \right\}$

由 Heston 模型描述, 即 ...

Finite-time ruin probability with an exponential Lévy process investment return and heavy-tailed claims

2009

... 同时, 根据文献 [15] 得到对任意固定的

$\delta>0$

$\alpha<p<\infty$

和足够大的

$x$

, 存在与

$Y$

和

$\delta$

无关的常数

$C>0$

使得下式成立 ...

On joint ruin probabilities of a two-dimensional risk model with constant interest rate

2013

Uniform asymptotic estimate for finite-time ruin probabilities of a time-dependent bidimensional renewal model

2015

A revisit to asymptotic ruin probabilities for a bidimensional renewal risk model

2018

The ruin probabilities of a discrete-time risk model with dependent insurance and financial risks

2016

Some asymptotic results of the ruin probabilities in a two-dimensional renewal risk model with some strongly subexponential claims

2016

Uniform estimate on finite time ruin probabilities with random interest rate

2010

2004

... 本证明的总体思路是首先计算

$E\big(\mathrm{e}^{-a\int_u^{u+v}r_sds}|\mathcal{F}_u\big)$

, 然后计算

$E\mathrm{e}^{-b\xi(u)-a\xi(u+v)}$

$r(t)$

是由随机微分方程 (3.2) 确定, 故

$r(t)$

是马尔科夫过程, 且对

$\widetilde{T}\geq t$

记

$B(t,\widetilde{T}\,):=E\big(\mathrm{e}^{-a\int_t^{\widetilde{T}}r_sds}|\mathcal{F}_t\big)=f\,(t,r_t)$

, 其中

$f\,(t,r)$

是关于虚拟变量

$t$

和

$r$

的函数. 为得到

$E\big(\mathrm{e}^{-a\int_u^{u+v}r_sds}|\mathcal{F}_u\big)$

, 先求解

$f\,(t,r)$

的显式表达式. 根据文献 [22], 需通过"建立鞅, 取微分, 令

$\mathrm{d}t$

项等于

$0$

" 这三步找到函数

$f\,(t,r)$

满足的偏微分方程, 通过求解偏微分方程得到

$f\,(t,r)$

的显式表达式. 对

$0\leq s\leq t \leq \widetilde{T}$

$E\left\{\mathrm{e}^{-a\xi(t)}\cdot B(t,\widetilde{T}\,)\Big|\mathcal{F}_s\right\} =E\left\{\mathrm{e}^{-a\int_0^tr_u\mathrm{d}u}\cdot E\big(\mathrm{e}^{-a\int_t^{\widetilde{T}}r_s\mathrm{d}s}|\mathcal{F}_t\big)|\mathcal{F}_s\right\} =E\left\{\mathrm{e}^{-a\int_0^{\widetilde{T}}r_u\mathrm{d}u}|\mathcal{F}_s\right\} =\mathrm{e}^{-a\xi(s)}\cdot B(s,\widetilde{T})$

. 故

$\mathrm{e}^{-a\xi(t)}\cdot B(t,\widetilde{T}\,)=\mathrm{e}^{-a\xi(t)}\cdot f\,(t,r_t)$

是鞅. 鞅

$\mathrm{e}^{-a\xi(t)}\cdot f\,(t,r_t)$

的微分为 ...

... 根据文献 [22], 该偏微分方程的解的形式为 ...

... 要得到

$E\left\{\mathrm{e}^{-C_1(u,u+v)r_u-(a+b)\xi(u)}\right\}$

, 只需考虑

$E\mathrm{e}^{p_1r_t+p_2\xi(t)}$

, 其中

$p_1$

$p_2$

是实数. 令

$D(p_1,p_2,t)=E\mathrm{e}^{p_1r_t+p_2\xi(t)}$

$r_t$

是高斯随机变量, 由文献 [22,例 4.4.10] 知

$\xi(t)=\int_0^t r_s \mathrm{d}s=(mlt+r_0-r_t+\delta W_t)/m$

也是高斯随机变量, 故对任意实数

$p_1$

和

$p_2$

$D(p_1,p_2,t)$

是有限的. 对

$\mathrm{e}^{p_1r_t+p_2\int_0^t r_s \mathrm{d}s}$

应用 Itô 公式并取期望得 ...

Asymptotic ruin probabilities of the renewal model with constant interest force and regular variation

2005

Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model

2010

... 设

$X$

和

$Y$

是两个独立的随机变量, 若

$X$

的分布函数

$F\in\mathcal{R}_{-\alpha}$

, 且对

$p>\alpha$

非负随机变量

$Y$

满足

$EY^{p}<\infty$

, 则由文献 [24] 的 (4.4) 式, 对任意给定的

$M\geq0$

有 ...

Asymptotic ruin probabilities for a bidimensional renewal risk model

2017

Tail behavior of the product of two dependent random variables with applications to risk theory

2012

... 根据文献^[26], 存在常数

$b_1$

$b_2$

和

$b_3$

使得 ...

On the first time of ruin in the bivariate compound Poisson model

2006

〈

〉