Asymptotics for Finite-Time Ruin Probabilities of a Dependent Bidimensional Risk Model with Stochastic Return and Subexponential Claims

Abstract

The paper considers a bidimensional continuous-time risk model with subexponential claims and Brownian perturbations, in which the price processes of the investment portfolio of the two lines of business are two geometric Lévy processes and the two lines of business share a common claim-number process, which is a renewal counting process. The paper mainly considers the claims of each line of business having a dependence structure. When the claims have subexponential distributions, the asymptotics of the finite-time ruin probabilities ${\psi }_{and}(x_1,x_2;T)$ and ${\psi }_{sim}(x_1,x_2;T)$ have been obtained. When the distributions of claims belong to the intersection of long-tailed and dominatedly varying-tailed distribution classes, the asymptotics of the finite-time ruin probability ${\psi }_{or}(x_1,x_2;T)$ is given.

1. Introduction

Consider a bidimensional continuous-time risk model with random return and Brownian perturbations. The discounted surplus process of $t\ge 0$ is expressed as

$$\begin{array}{ccc}\hfill \left(\begin{array}{c}{U}_1\left(t\right)\\ U_2\left(t\right)\end{array}\right)& =& \left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)+\left(\begin{array}{c}{\int }_{0-}^t{e}^{-R_1\left(s\right)}{C}_1\left(ds\right)\\ {\int }_{0-}^t{e}^{-R_2\left(s\right)}{C}_2\left(ds\right)\end{array}\right)+\left(\begin{array}{c}{\delta }_1{\int }_{0-}^t{e}^{-{\tilde{R}}_1\left(s\right)}{B}_1\left(ds\right)\\ {\delta }_2{\int }_{0-}^t{e}^{-{\tilde{R}}_2\left(s\right)}{B}_2\left(ds\right)\end{array}\right)\hfill \\ & & -\left(\begin{array}{c}{\displaystyle \sum _{i=1}^{N\left(t\right)}}{X}_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}\\ {\displaystyle \sum _{j=1}^{N\left(t\right)}}{X}_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_j\right)}\end{array}\right),\hfill \end{array}$$

(1)

where $(x_1,x_2)$ denotes the vector of the initial surpluses, $\left\{\left(X_k^{\left(1\right)},X_k^{\left(2\right)}\right);k\ge 1\right\}$ represents the sequence of identically distributed claim vectors with marginal distributions $F_1$ and $F_2$ on $\left[0,\infty \right)$, respectively. For the ith ($i=1,2$) line of business, $C_i\left(t\right)={\int }_0^t{c}_i\left(s\right)ds$ denotes the premium accumulation up to time $t\ge 0$; here, $c_i\left(t\right)$ is the density function of premium income of the ith line of business at time $t\ge 0$. Assume that the densities $c_i\left(t\right),i=1,2,$ are bounded, i.e., $0\le c_i\left(t\right)\le M$ for any $t\ge 0,i=1,2,$ and some constant $M>0$. $\left\{e^{R_i\left(t\right)};t\ge 0\right\}$ denotes the price process of the investment portfolio of the ith $(i=1,2)$ line of business; here, $\left\{R_i\left(t\right);t\ge 0\right\},i=1,2,$ are two nonnegative Lévy processes which start from 0 and have independent and stationary increments. $\left\{B_i\left(t\right);t\ge 0\right\}$ is the standard Brownian motion with the volatility coefficient ${\delta }_i\ge 0,i=1,2$. $\left\{{\tilde{R}}_i\left(t\right);t\ge 0\right\},i=1,2,$ are other nonnegative Lévy processes which also start from 0 and have independent and stationary increments. The properties of the Lévy process can be found in the monograph [1]. $\left\{{\sigma }_k;k\ge 1\right\}$ is the sequence of the claim-arrival times, which constitutes the claim-number process $\left\{N\left(t\right);t\ge 0\right\}$ with $N\left(t\right)=sup\left\{k\ge 1:{\sigma }_k\le t\right\},t\ge 0$, and the claim-number process $\left\{N\left(t\right);t\ge 0\right\}$ is a renewal counting process. Let ${\theta }_j={\sigma }_j-{\sigma }_{j-1},j\ge 2$, and ${\theta }_1={\sigma }_1$ be the inter-arrival times of claims, which are independent and identically distributed (i.i.d.) random variables. Denote the mean function by

$$\begin{array}{c}\hfill \lambda \left(t\right)=E\left[N\left(t\right)\right]={\sum }_{k=1}^{\infty }P\left({\sigma }_k\le t\right),t\ge 0\end{array}$$

and define $\mathsf{\Lambda }=\left\{t>0:\lambda \left(t\right)>0\right\}$. We assume that $\left\{X_k^{\left(1\right)};k\ge 1\right\}$, $\left\{X_k^{\left(2\right)};k\ge 1\right\}$, $\{B_1\left(t\right);t\ge 0\}$, $\left\{B_2\left(t\right);t\ge 0\right\}$, $\left\{{\theta }_j;j\ge 1\right\}$, $\left\{R_1\left(t\right);t\ge 0\right\}$, $\left\{R_2\left(t\right);t\ge 0\right\}$, $\{{\tilde{R}}_1\left(t\right);t\ge 0\}$ and $\{{\tilde{R}}_2\left(t\right);$$t\ge 0\}$ are mutually independent.

For the above bidimensional risk model (1), we do not consider the case that the surplus of one line is used to cover a deficit of another line. Chan et al. [2], Li et al. [3], Chen et al. [4] and Cheng et al. [5] define three types of finite-time ruin probabilities for any time $T\ge 0$ as

$$\begin{array}{ccc}\hfill {\psi }_{sim}\left(x_1,x_2;T\right)& =& P\left(\underset{0\le t\le T}{inf}\left\{U_1\left(t\right)\vee U_2\left(t\right)\right\}<0\right)\hfill \\ & =& P\left(U_1\left(t\right)<0\mathrm{and}{U}_2\left(t\right)<0\mathrm{for} \mathrm{some}0\le t\le T\right),\hfill \end{array}$$

$$\begin{array}{c}\hfill {\psi }_{and}\left(x_1,x_2;T\right)=P\left(\underset{0\le t\le T}{inf}{U}_1\left(t\right)<0\mathrm{and}\underset{0\le t\le T}{inf}{U}_2\left(t\right)<0\right)\end{array}$$

and

$$\begin{array}{c}\hfill {\psi }_{or}\left(x_1,x_2;T\right)=P\left(\underset{0\le t\le T}{inf}{U}_1\left(t\right)<0\mathrm{or}\underset{0\le t\le T}{inf}{U}_2\left(t\right)<0\right),\end{array}$$

where $U_1\left(t\right)\vee U_2\left(t\right)=max\left\{U_1\left(t\right),U_2\left(t\right)\right\},t\ge 0$. Clearly, for any $T\ge 0$ and $x_1,x_2\ge 0$, it holds obviously that

$$\begin{array}{c}\hfill {\psi }_{sim}\left(x_1,x_2;T\right)\le {\psi }_{and}\left(x_1,x_2;T\right)\le {\psi }_{or}\left(x_1,x_2;T\right).\end{array}$$

In the past decade, researchers have paid more attention to multidimensional risk models, especially the bidimensional risk model, and investigated the asymptotic behavior of ruin probabilities. One can refer to [6,7,8], and so on. Recently, Cheng et al. [9] considered a bidimensional risk model in which the price process of the investment portfolio is a constant force of interest and the claims of each line of business have a dependence structure (i.e., Assumption 2 below). They obtained asymptotics of ${\psi }_{sim}\left(x_1,x_2;T\right)$ and ${\psi }_{and}\left(x_1,x_2;T\right)$ for subexponential claims. However, when the insurance companies invest their assets into a financial market, the return on the investment will be random. Many researchers have considered the price process of the investment portfolio is a geometric Lévy process, such as [10,11], and so on. Xu et al. [12] considered a one-dimensional renewal risk model with stochastic return and dependent claims satisfying Assumption 2 below.

Inspired by the works of Cheng et al. [9] and Xu et al. [12], the paper mainly studies the asymptotics for ${\psi }_{sim}\left(x_1,x_2;T\right)$, ${\psi }_{and}\left(x_1,x_2;T\right)$ and ${\psi }_{or}\left(x_1,x_2;T\right)$ as $\left(x_1,x_2\right)\to \left(\infty ,\infty \right)$ in the risk model (1), where the price processes of the investment portfolio of two lines of business are two geometric Lévy processes, and the claims of two lines of business have a dependence structure (i.e., Assumption 2 below), respectively.

The rest of the paper is organized as follows. Section 2 introduces some preliminaries, including heavy-tailed distribution classes and some dependence structures. Section 3 provides the main results of this paper. In Section 4, we give some lemmas and present the proofs of the main results.

2. Preliminaries

Throughout this article, all limit relationships hold as $x\to \infty$ or $\left(x_1,x_2\right)\to \left(\infty ,\infty \right)$, corresponding to univariate or bivariate cases, respectively, unless otherwise specified. For two univariate or bivariate functions $f(·)$ and $g(·)$, write $f=O\left(g\right)$ if $lim\; supf/g<\infty$; write $f=o\left(g\right)$ if $limf/g=0$; write $f\asymp g$ if both $f=O\left(g\right)$ and $g=O\left(f\right)$; write $f\lesssim g$ if $limsupf/g\le 1$; write $f\gtrsim g$ if $liminff/g\ge 1$; and write $f\sim g$ if $limf/g=1$.

In the following, we will present the definition of a heavy-tailed distribution and a few important subclasses of a heavy-tailed distribution class. Let V be a proper distribution and for any $x\in \left(-\infty ,\infty \right)$, we denote its tail by $\overline{V}\left(x\right)=1-V\left(x\right)$.

A distribution V is said to be heavy-tailed on $\left(-\infty ,\infty \right)$, if for any $\lambda >0$,

$$\begin{array}{c}\hfill {\int }_{-\infty }^{\infty }{e}^{\lambda x}V\left(dx\right)=\infty .\end{array}$$

A distribution V supported on $\left[0,\infty \right)$ is said to belong to the subexponential distribution class, denoted by $V\in \mathcal{S}$, if

$$\begin{array}{c}\hfill \underset{x\to \infty }{lim}{\displaystyle \frac{\overline{V^{\ast n}}\left(x\right)}{\overline{V}\left(x\right)}}=n\end{array}$$

holds for some (or, equivalently, for all) $n\ge 2$, where $V^{\ast n}$ denotes the n-fold convolution of V, $n\ge 2$.

A distribution V supported on $\left(-\infty ,\infty \right)$ is said to belong to the long-tailed distribution class, denoted by $V\in \mathcal{L}$, if for any $u\in (-\infty ,\infty )$, it holds that

$$\begin{array}{c}\hfill \underset{x\to \infty }{lim}{\displaystyle \frac{\overline{V}\left(x+u\right)}{\overline{V}\left(x\right)}}=1.\end{array}$$

Another important subclass of a heavy-tailed distribution class is the dominatedly varying-tailed distribution class. A distribution V supported on $\left(-\infty ,\infty \right)$ is said to belong to the dominatedly varying-tailed distribution class, denoted by $V\in \mathcal{D}$, if for any $0<u<1$,

$$\begin{array}{c}\hfill \underset{x\to \infty }{limsup}{\displaystyle \frac{\overline{V}\left(xu\right)}{\overline{V}\left(x\right)}}<\infty .\end{array}$$

For the aforementioned heavy-tailed distribution subclasses, the following inclusion relations hold:

$$\begin{array}{c}\hfill \mathcal{L}\cap \mathcal{D}\subset \mathcal{S}\subset \mathcal{L},\end{array}$$

see, e.g., [13,14,15].

In many previous studies of risk theory, various of dependence structures have been proposed. In the following, we will use the dependence structure introduced by Ko and Tang [16].

Assumption 1.

Let ${\eta }_i,1\le i\le n$, be n nonnegative random variables. There exist some constant $x_0=x_0\left(n\right)>0$ and a dominating coefficient $d_n>0$ such that for any $2\le j\le n$,

$$\begin{array}{c}\hfill {\displaystyle \frac{P\left({\sum }_{i=1}^{j-1}{\eta }_i>x-t|{\eta }_j=t\right)}{P\left({\sum }_{i=1}^{j-1}{\eta }_i>x-t\right)}}\le d_n\end{array}$$

holds uniformly for all $x>x_0$ and $t\in \left[x_0,x\right]$.

For Assumption 1, Jiang et al. [17] extended the supports of distributions from $\left[0,\infty \right)$ to $\left(-\infty ,\infty \right)$. Yang et al. [18] modified Assumption 1 as follows to model the dependence structure of a sequence of claims in risk theory.

Assumption 2.

Let ${\eta }_i,i\ge 1$, be a sequence of nonnegative random variables. There exist some constant $x_0>0$ and a dominating coefficient $d>0$ such that for any $n\ge 2$ ($x_0$ and d are irrespective of n),

$$\begin{array}{c}\hfill {\displaystyle \frac{P\left({\sum }_{i=1}^{n-1}{\eta }_i>x-t|{\eta }_n=t\right)}{P\left({\sum }_{i=1}^{n-1}{\eta }_i>x-t\right)}}\le d\end{array}$$

holds uniformly for all $x>x_0$ and $t\in \left[x_0,x\right]$.

3. Main Results

In the following, we present the main results of the paper.

Theorem 1.

Consider the bidimensional risk model (1). Assume that $\left\{X_k^{\left(1\right)};k\ge 1\right\}$ and $\left\{X_k^{\left(2\right)};k\ge 1\right\}$ satisfy Assumption 2 with a dominating coefficient d, respectively. If $F_i\in \mathcal{S},i=1,2$, then for any fixed $T\in \mathsf{\Lambda }$, it holds that

$$\begin{array}{ccc}& & {\psi }_{and}\left(x_1,x_2;T\right)\sim {\psi }_{sim}\left(x_1,x_2;T\right)\hfill \\ & \sim & \underset{s,u\ge 0,s+u\le T}{\int \int }(P\left(X_1^{\left(1\right)}{e}^{-R_1(u+s)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\hfill \\ & & +P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2(u+s)}>x_2\right))\lambda \left(du\right)\lambda \left(ds\right)\hfill \\ & & +{\int }_{0-}^{T}P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\lambda \left(du\right).\hfill \end{array}$$

(2)

Remark 1.

(1) When $R_i\left(t\right)=rt$, for some constant $r\ge 0$, any $t\ge 0$ and $i=1,2$, the result (2) coincides with (2.2) of [19] with $\theta =0$ (the case of independent claims).

(2) If $R_i\left(t\right)=0$ for all $t\ge 0$ and $i=1,2$ and $\left\{N\right(t);t\ge 0\}$ is a Poisson process with intensity $\lambda >0$, then by (2), it holds that

$$\begin{array}{c}\hfill {\psi }_{and}\left(x_1,x_2;T\right)\sim {\psi }_{sim}\left(x_1,x_2;T\right)\sim \lambda T(\lambda T+1)\overline{F_1}\left(x_1\right)\overline{F_2}\left(x_2\right),\end{array}$$

which is the result of Theorem 4.1 of Li et al. [3].

(3) Theorem 1 shows that in the bidimensional risk model with stochastic return, the asymptotics of the finite-time ruin probabilities are insensitive to the Brownian perturbation when the claims have heavy-tailed distributions.

Particularly, if the distributions of claims belong to the class $\mathcal{L}\cap \mathcal{D}$, then the asymptotics of finite-time ruin probability ${\psi }_{or}(x_1,x_2;T)$ are given.

Theorem 2.

Under the conditions of Theorem 1, if $F_i\in \mathcal{L}\cap \mathcal{D},i=1,2$, then for any fixed $T\in \mathsf{\Lambda }$, it holds that

$$\begin{array}{c}\hfill {\psi }_{or}\left(x_1,x_2;T\right)\sim {\int }_{0-}^{T}P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)\lambda \left(du\right)+{\int }_{0-}^{T}P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\lambda \left(du\right).\end{array}$$

(3)

4. Proofs of Main Results

Before proving the main results of the paper, we first give a series of lemmas. The first lemma is from Lemma 3.1 of [12].

Lemma 1.

Let ${\eta }_i,1\le i\le n$ be n nonnegative random variables with distributions of $V_i\in \mathcal{L},1\le i\le n$, respectively. ${\eta }_0$ is a real-valued random variable with a distribution of $V_0$, which is independent of ${\eta }_i,1\le i\le n$. Assume that ${\eta }_i,1\le i\le n$ satisfy Assumption 1 with a dominating coefficient $d_n$ and $\overline{V_i}\left(x\right)\asymp \overline{V}\left(x\right)$ for some $V\in \mathcal{S}$ and all $1\le i\le n$. The random weights ${\theta }_i,1\le i\le n$ are nonnegative, not degenerate at zero and arbitrarily dependent on each other, but independent of ${\eta }_i,0\le i\le n$. If $\overline{V_0}\left(x\right)=o\left(\overline{V}\left(x/c\right)\right)$ for all $c>0$ and ${\theta }_i,1\le i\le n$ are bounded above, then

$$\begin{array}{c}\hfill P\left(\sum _{i=1}^n{\theta }_i{\eta }_i+{\eta }_0>x\right)\sim \sum _{i=1}^{n}P\left({\theta }_i{\eta }_i>x\right).\end{array}$$

The next lemma is from Lemma 3.2 of [12].

Lemma 2.

Let ${\eta }_i,i\ge 1$ be a sequence of nonnegative random variables satisfying Assumption 2 with a dominating coefficient d and a common distribution $V\in \mathcal{S}$. ${\eta }_0$ is a real-valued random variable with a distribution of $V_0$ and the random variable θ is nonnegative, upper bounded and not degenerate at zero. Assume that $\{{\eta }_i,i\ge 1\}$, ${\eta }_0$ and θ are independent, G is the distribution of $\theta {\eta }_1$ and $\overline{V_0}\left(x\right)=O\left(\overline{G}\left(x\right)\right)$. Then for any $\epsilon >0$, there exists some constant $K=K(\epsilon ,V,V_0,G)>0$ such that for all $x\in \left(-\infty ,\infty \right)$ and $n\ge 1$,

$$\begin{array}{c}\hfill P\left(\sum _{i=1}^n\theta {\eta }_i+{\eta }_0>x\right)\le K{\left(1+d\epsilon \right)}^{n}P\left(\theta {\eta }_1>x\right).\end{array}$$

In the risk model (1), for any $T\ge 0$ and $i=1,2$, suppose that

$$p_i\left(T\right)={\int }_{0-}^T{e}^{-{\tilde{R}}_i\left(s\right)}{B}_i\left(ds\right)$$

and set

$$p_{i\ast }(T)=\underset{0\le s\le T}{inf}{p}_i\left(s\right)\le 0\mathrm{and}{p}_i^{*}\left(T\right)=\underset{0\le s\le T}{sup}{p}_i\left(s\right)\ge 0.$$

The result of the following lemma is (19) of [20].

Lemma 3.

Consider the risk model (1) with the nonnegative Lévy processes of $\left\{{\tilde{R}}_i\left(t\right);t\ge 0\right\},i=1,2$. Then for any $T>0$ and $x>0$, it holds that

$$\begin{array}{c}\hfill P\left({\delta }_i{p}_{i\ast }\left(T\right)<-x\right)=P\left({\delta }_i{p}_i^{*}\left(T\right)>x\right)\le 2\overline{\mathsf{\Phi }}\left({\displaystyle \frac{x}{{\delta }_i\sqrt{T}}}\right),i=1,2,\end{array}$$

where Φ is the standard Gaussian distribution and ${\displaystyle \frac{1}{{\delta }_i}}$ is understood as ∞ when ${\delta }_i=0,i=1,2$.

Proof of Theorem 1.

First of all, we prove the lower bound of ${\psi }_{and}\left(x_1,x_2;T\right)$, i.e., for any fixed $T\in \mathsf{\Lambda }$,

$$\begin{array}{ccc}& & {\psi }_{and}(x_1,x_2;T)\hfill \\ & \gtrsim & \underset{s,u\ge 0,s+u\le T}{\int \int }(P\left(X_1^{\left(1\right)}{e}^{-R_1(u+s)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\hfill \\ & & +P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2(u+s)}>x_2\right))\lambda \left(du\right)\lambda \left(ds\right)\hfill \\ & & +{\int }_{0-}^{T}P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\lambda \left(du\right).\hfill \end{array}$$

(4)

For any $T>0$ and $n\ge 1$, setting

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_{\left(n\right)}=\left\{\left(y_1,y_2,\cdots ,y_{n+1}\right):0\le y_1\le \cdots \le y_n\le T<y_{n+1}\right\}.\end{array}$$

Since for any $T>0$ and $i=1,2$, $\{U_i\left(T\right)<0\}\subset \{{inf}_{0\le t\le T}{U}_i\left(t\right)<0\}$, $p_i\left(T\right)\le p_i^{*}\left(T\right)$ and

$${\int }_{0-}^T{e}^{-R_i\left(s\right)}{C}_i\left(ds\right)={\int }_0^T{c}_i\left(s\right)e^{-R_i\left(s\right)}ds\le {\int }_0^{T}Mds=MT,$$

it follows from the definition of ${\psi }_{and}(x_1,x_2;T)$ that for any $T>0$ and $x_1,x_2>0$,

$$\begin{array}{ccc}& & {\psi }_{and}(x_1,x_2;T)\hfill \\ & \ge & P(U_1\left(T\right)<0,U_2\left(T\right)<0)\hfill \\ & =& P(\sum _{i=1}^{N\left(T\right)}{X}_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}-{\int }_{0-}^T{e}^{-R_1\left(s\right)}{C}_1\left(ds\right)-{\delta }_1{p}_1\left(T\right)>x_1,\hfill \\ & & \sum _{j=1}^{N\left(T\right)}{X}_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_j\right)}-{\int }_{0-}^T{e}^{-R_2\left(s\right)}{C}_2\left(ds\right)-{\delta }_2{p}_2\left(T\right)>x_2)\hfill \\ & \ge & P\left(\sum _{i=1}^{N\left(T\right)}{X}_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}-{\delta }_1{p}_1^{*}\left(T\right)-MT>x_1,\sum _{j=1}^{N\left(T\right)}{X}_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_j\right)}-{\delta }_2{p}_2^{*}\left(T\right)-MT>x_2\right).\hfill \end{array}$$

Hence, for any integer $N_1>0$, $x_1,x_2>0$ and $T\in \mathsf{\Lambda }$, it holds that

$$\begin{array}{ccc}& & {\psi }_{and}(x_1,x_2;T)\hfill \\ & \ge & \sum _{n=1}^{\infty }P(\sum _{i=1}^n{X}_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}-{\delta }_1{p}_1^{*}\left(T\right)-MT>x_1,\hfill \\ & & \sum _{j=1}^n{X}_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_j\right)}-{\delta }_2{p}_2^{*}\left(T\right)-MT>x_2,N\left(T\right)=n)\hfill \\ & \ge & \sum _{n=1}^{N_1}P(\sum _{i=1}^n{X}_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}-{\delta }_1{p}_1^{*}\left(T\right)-MT>x_1,\hfill \\ & & \sum _{j=1}^n{X}_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_j\right)}-{\delta }_2{p}_2^{*}\left(T\right)-MT>x_2,{\sigma }_i\le T,1\le i\le n,{\sigma }_{n+1}>T)\hfill \\ & =& \sum _{n=1}^{N_1}\int \cdots \underset{\mathsf{\Omega }\left(n\right)}{\int }P(\sum _{i=1}^n{X}_i^{\left(1\right)}{e}^{-R_1\left(y_i\right)}-{\delta }_1{p}_1^{*}\left(T\right)-MT>x_1,\hfill \\ & & \sum _{j=1}^n{X}_j^{\left(2\right)}{e}^{-R_2\left(y_j\right)}-{\delta }_2{p}_2^{*}\left(T\right)-MT>x_2)P\left({\sigma }_1\in d{y}_1,\cdots ,{\sigma }_{n+1}\in d{y}_{n+1}\right)\hfill \\ & =& \sum _{n=1}^{N_1}\int \cdots \underset{\mathsf{\Omega }\left(n\right)}{\int }P\left(\sum _{i=1}^n{X}_i^{\left(1\right)}{e}^{-R_1\left(y_i\right)}-{\delta }_1{p}_1^{*}\left(T\right)-MT>x_1\right)\hfill \\ & & P\left(\sum _{j=1}^n{X}_j^{\left(2\right)}{e}^{-R_2\left(y_j\right)}-{\delta }_2{p}_2^{*}\left(T\right)-MT>x_2\right)P\left({\sigma }_1\in d{y}_1,\cdots ,{\sigma }_{n+1}\in d{y}_{n+1}\right).\hfill \end{array}$$

(5)

For any $T>0$, $-p_i^{*}\left(T\right)-MT<0$, then, for all $c>0$, $x>0$ and $i=1,2$, we have

$$\begin{array}{c}\hfill P\left(-{\delta }_i{p}_i^{*}\left(T\right)-MT>x\right)=o\left({\overline{F}}_i\left(x/c\right)\right).\end{array}$$

Thus, by Lemma 1 and (5), we can get that for any fixed $T\in \mathsf{\Lambda }$,

$$\begin{array}{ccc}& & {\psi }_{and}\left(x_1,x_2;T\right)\hfill \\ & \gtrsim & \sum _{n=1}^{N_1}\sum _{i=1}^n\sum _{j=1}^n\int \cdots \underset{\mathsf{\Omega }\left(n\right)}{\int }P\left(X_i^{\left(1\right)}{e}^{-R_1\left(y_i\right)}>x_1\right)P\left(X_j^{\left(2\right)}{e}^{-R_2\left(y_j\right)}>x_2\right)\hfill \\ & & P\left({\sigma }_1\in d{y}_1,\cdots ,{\sigma }_{n+1}\in d{y}_{n+1}\right)\hfill \\ & =& \sum _{n=1}^{N_1}\sum _{i=1}^n\sum _{j=1}^{n}P\left(X_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}>x_1,X_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_j\right)}>x_2,N\left(T\right)=n\right)\hfill \\ & =& \left(\sum _{n=1}^{\infty }-\sum _{n=N_1+1}^{\infty }\right)\sum _{i=1}^n\sum _{j=1}^{n}P\left(X_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}>x_1,X_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_j\right)}>x_2,N\left(T\right)=n\right)\hfill \\ & =:& Q_1(x_1,x_2;T)-Q_2(x_1,x_2;T).\hfill \end{array}$$

(6)

We use the method of the proof of Lemma 3.4 in [19] to deal with $Q_1(x_1,x_2;T)$. By exchanging the order of sums, it holds for any $x_1,x_2>0$ and $T\in \mathsf{\Lambda }$ that

$$\begin{array}{ccc}& & Q_1(x_1,x_2;T)\hfill \\ & =& \sum _{i=1}^{\infty }\sum _{n=i}^{\infty }\sum _{j=1}^{n}P\left(X_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}>x_1,X_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_j\right)}>x_2,N\left(T\right)=n\right)\hfill \\ & =& \sum _{i=1}^{\infty }\sum _{n=i}^{\infty }\left(\sum _{j=1}^{i-1}+\sum _{j=i}+\sum _{j=i+1}^n\right)P\left(X_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}>x_1,X_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_j\right)}>x_2,N\left(T\right)=n\right)\hfill \\ & =& \sum _{j=1}^{\infty }\sum _{i=j+1}^{\infty }P\left(X_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}>x_1,X_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_j\right)}>x_2,{\sigma }_i\le T\right)\hfill \\ & & +\sum _{i=1}^{\infty }P\left(X_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}>x_1,X_i^{\left(2\right)}{e}^{-R_2\left({\sigma }_i\right)}>x_2,{\sigma }_i\le T\right)\hfill \\ & & +\sum _{i=1}^{\infty }\sum _{j=i+1}^{\infty }P\left(X_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}>x_1,X_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_j\right)}>x_2,{\sigma }_j\le T\right)\hfill \\ & =:& J_1\left(x_1,x_2;T\right)+J_2\left(x_1,x_2;T\right)+J_3\left(x_1,x_2;T\right).\hfill \end{array}$$

(7)

For $J_1(x_1,x_2;T)$, since $\left\{N\right(t);t\ge 0\}$ is a renewal counting process, for any $i>j\ge 1$, ${\sigma }_i-{\sigma }_j$ and ${\sigma }_j$ are independent and ${\sigma }_i-{\sigma }_j$ has the same distribution as ${\sigma }_{i-j}$. Then, for any $x_1,x_2>0$ and $T\in \mathsf{\Lambda }$,

$$\begin{array}{ccc}& & J_1(x_1,x_2;T)\hfill \\ & =& \sum _{j=1}^{\infty }\sum _{i=j+1}^{\infty }P\left(X_i^{\left(1\right)}{e}^{-R_1({\sigma }_j+({\sigma }_i-{\sigma }_j))}>x_1,X_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_j\right)}>x_2,{\sigma }_j+({\sigma }_i-{\sigma }_j)\le T\right)\hfill \\ & =& \sum _{j=1}^{\infty }\sum _{i=j+1}^{\infty }\underset{s,u\ge 0,s+u\le T}{\int \int }P\left(X_1^{\left(1\right)}{e}^{-R_1(u+s)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)P\left({\sigma }_{i-j}\in ds\right)P\left({\sigma }_j\in du\right)\hfill \\ & =& \underset{s,u\ge 0,s+u\le T}{\int \int }P\left(X_1^{\left(1\right)}{e}^{-R_1(u+s)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\lambda \left(ds\right)\lambda \left(du\right).\hfill \end{array}$$

(8)

Similarly, we can get for any $x_1,x_2>0$ and $T\in \mathsf{\Lambda }$,

$$\begin{array}{c}\hfill J_3(x_1,x_2;T)=\underset{s,u\ge 0,s+u\le T}{\int \int }P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2(u+s)}>x_2\right)\lambda \left(ds\right)\lambda \left(du\right).\end{array}$$

(9)

For $J_2(x_1,x_2;T)$, it holds for any $x_1,x_2>0$ and $T\in \mathsf{\Lambda }$ that

$$\begin{array}{ccc}\hfill J_2(x_1,x_2;T)& =& \sum _{i=1}^{\infty }{\int }_{0-}^{T}P\left(X_i^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1,X_i^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)P({\sigma }_i\in du)\hfill \\ & =& {\int }_{0-}^{T}P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\lambda \left(du\right),\hfill \end{array}$$

which, combining with (7)–(9), yields that

$$\begin{array}{ccc}& & Q_1(x_1,x_2;t)\hfill \\ & =& \underset{s,u\ge 0,s+u\le T}{\int \int }P\left(X_1^{\left(1\right)}{e}^{-R_1(u+s)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\lambda \left(ds\right)\lambda \left(du\right)\hfill \\ & & +\underset{s,u\ge 0,s+u\le T}{\int \int }P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2(u+s)}>x_2\right)\lambda \left(ds\right)\lambda \left(du\right)\hfill \\ & & +{\int }_{0-}^{T}P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\lambda \left(du\right).\hfill \end{array}$$

(10)

In the following, we deal with $Q_2(x_1,x_2;T)$. Since $\{{\theta }_i;i\ge 1\}$ are i.i.d. random variables, for any $x_1,x_2>0$ and $T\in \mathsf{\Lambda }$, it holds that

$$\begin{array}{ccc}& & Q_2\left(x_1,x_2;T\right)\hfill \\ & \le & \sum _{n=N_1+1}^{\infty }\sum _{i=1}^n\sum _{j=1}^{n}P\left(X_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_1\right)}>x_1,X_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_1\right)}>x_2,{\sigma }_n\le T\right)\hfill \\ & =& \sum _{n=N_1+1}^{\infty }\sum _{i=1}^n\sum _{j=1}^n{\int }_{0-}^{T}P\left(X_i^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1,X_j^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2,\sum _{k=2}^n{\theta }_k\le T-u\right)P({\sigma }_1\in du)\hfill \\ & =& \sum _{n=N_1+1}^{\infty }\sum _{i=1}^n\sum _{j=1}^n{\int }_{0-}^{T}P\left(X_i^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1,X_j^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2,N(T-u)\ge n-1\right)P({\sigma }_1\in du)\hfill \\ & \le & \sum _{n=N_1+1}^{\infty }{n}^2{\int }_{0-}^{T}P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)P\left(N\left(T\right)\ge n-1\right)P\left({\sigma }_1\in du\right)\hfill \\ & \le & {\int }_{0-}^{T}P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\lambda \left(du\right)\sum _{n=N_1+1}^{\infty }{n}^{2}P(N\left(T\right)\ge n-1).\hfill \end{array}$$

Since $\left\{N\right(t);t\ge 0\}$ is a renewal counting process, it holds that

$$\underset{N_1\to \infty }{lim}\sum _{n=N_1+1}^{\infty }{n}^{2}P(N\left(T\right)\ge n-1)=0.$$

Thus, letting first $(x_1,x_2)\to (\infty ,\infty )$ and then $N_1\to \infty$, it holds that

$$\begin{array}{ccc}\hfill Q_2\left(x_1,x_2;T\right)& =& o\left({\int }_{0-}^{T}P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\lambda \left(du\right)\right)\hfill \\ & =& o\left(Q_1(x_1,x_2;T)\right).\hfill \end{array}$$

(11)

By (6), (10) and (11), we get that (4) holds.

In the following, we turn to estimate the upper bound of ${\psi }_{and}\left(x_1,x_2;T\right)$. Choosing any integer $N_1>0$, by Lemma 3, for any $x_1,x_2>0$ and $T\in \mathsf{\Lambda }$,

$$\begin{array}{ccc}& & {\psi }_{and}(x_1,x_2;T)\hfill \\ & \le & P\left(\sum _{i=1}^{N\left(T\right)}{X}_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}-{\delta }_1{p}_{1\ast }\left(T\right)>x_1,\sum _{j=1}^{N\left(T\right)}{X}_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_j\right)}-{\delta }_2{p}_{2\ast }\left(T\right)>x_2\right)\hfill \\ & =& \sum _{n=1}^{\infty }P\left(\sum _{i=1}^n{X}_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}+{\delta }_1{p}_1^{*}\left(T\right)>x_1,\sum _{j=1}^n{X}_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_j\right)}+{\delta }_2{p}_2^{*}\left(T\right)>x_2,N\left(T\right)=n\right)\hfill \\ & =& \left(\sum _{n=1}^{N_1}+\sum _{n=N_1+1}^{\infty }\right)\hfill \\ & & P\left(\sum _{i=1}^n{X}_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}+{\delta }_1{p}_1^{*}\left(T\right)>x_1,\sum _{j=1}^n{X}_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_j\right)}+{\delta }_2{p}_2^{*}\left(T\right)>x_2,N\left(T\right)=n\right)\hfill \\ & =:& I_1(x_1,x_2;T)+I_2(x_1,x_2;T).\hfill \end{array}$$

(12)

For $I_2(x_1,x_2;T)$, since $F_i\in \mathcal{S},i=1,2$, and $\{R_i\left(t\right);t\ge 0\},i=1,2$, are nonnegative Lévy processes, it follows from Corollary 2.5 of [21] that $X_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}$ and $X_i^{\left(2\right)}{e}^{-R_2\left({\sigma }_i\right)}$ have subexponential distributions for all $i\ge 1$. Thus, by Lemma 3 for $k=1,2$, any $T>0$ and $i\ge 1$,

$$P\left({\delta }_k{p}_k^{*}\left(T\right)>x\right)=o\left(P\left(X_i^{\left(k\right)}{e}^{-R_k\left({\sigma }_i\right)}>x\right)\right).$$

By Lemma 2, for any $\epsilon >0$, there exists a constant $K>0$ such that for any $x_1,x_2>0$ and $T\in \mathsf{\Lambda }$,

$$\begin{array}{ccc}& & I_2(x_1,x_2;T)\hfill \\ & \le & \sum _{n=N_1+1}^{\infty }{\int }_{0-}^{T}P\left(\sum _{i=1}^n{X}_i^{\left(1\right)}{e}^{-R_1\left(y_1\right)}+{\delta }_1{p}_1^{*}\left(T\right)>x_1\right)P\left(\sum _{j=1}^n{X}_j^{\left(2\right)}{e}^{-R_2\left(y_1\right)}+{\delta }_2{p}_2^{*}\left(T\right)>x_2\right)\hfill \\ & & P(N\left(t\right)\ge n-1)P({\sigma }_1\in d{y}_1)\hfill \\ & \le & \sum _{n=N_1+1}^{\infty }{K}^2{(1+d\epsilon )}^{2n}P(N\left(t\right)\ge n-1)\hfill \\ & & {\int }_{0-}^{T}P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\lambda \left(du\right).\hfill \end{array}$$

Since $\left\{N\right(t);t\ge 0\}$ is a renewal counting process, it holds for some small constant $\epsilon >0$ that

$$\underset{N_1\to \infty }{lim}\sum _{n=N_1+1}^{\infty }{\left(1+d\epsilon \right)}^{2n}P(N\left(T\right)\ge n-1)=0.$$

Therefore, first letting $(x_1,x_2)\to (\infty ,\infty )$ and then $N_1\to \infty$, it holds for any $T\in \mathsf{\Lambda }$ that

$$\begin{array}{c}\hfill I_2(x_1,x_2;T)=o\left({\int }_{0-}^{T}P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\lambda \left(du\right)\right).\end{array}$$

(13)

For $I_1(x_1,x_2;T)$, using a similar method of the proof of (6), by Lemma 1 and (10), first letting $(x_1,x_2)\to (\infty ,\infty )$ and then letting $N_1\to \infty$, it holds that

$$\begin{array}{ccc}& & I_1(x_1,x_2;T)\hfill \\ & \sim & \sum _{n=1}^{\infty }\sum _{i=1}^n\sum _{j=1}^{n}P\left(X_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}>x_1,X_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_j\right)}>x_2,N\left(T\right)=n\right)\hfill \\ & =& \underset{s,u\ge 0,s+u\le T}{\int \int }P\left(X_1^{\left(1\right)}{e}^{-R_1(u+s)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\lambda \left(ds\right)\lambda \left(du\right)\hfill \\ & & +\underset{s,u\ge 0,s+u\le T}{\int \int }P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2(u+s)}>x_2\right)\lambda \left(ds\right)\lambda \left(du\right)\hfill \\ & & +{\int }_{0-}^{T}P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\lambda \left(du\right).\hfill \end{array}$$

(14)

Therefore, by (12)–(14), it holds that

$$\begin{array}{ccc}& & {\psi }_{and}\left(x_1,x_2;T\right)\hfill \\ & \lesssim & \underset{s,u\ge 0,s+u\le T}{\int \int }(P\left(X_1^{\left(1\right)}{e}^{-R_1(u+s)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\hfill \\ & & +P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2(u+s)}>x_2\right))\lambda \left(du\right)\lambda \left(ds\right)\hfill \\ & & +{\int }_{0-}^{T}P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\lambda \left(du\right),\hfill \end{array}$$

(15)

which, combined with (4), yields the asymptotic result of ${\psi }_{and}(x_1,x_2;T)$.

In the following, we prove the result of ${\psi }_{sim}(x_1,x_2;T)$. By the same method of handling the lower bound of ${\psi }_{and}(x_1,x_2;T)$, for any $T\in \mathsf{\Lambda }$, it holds that

$$\begin{array}{ccc}& & {\psi }_{sim}(x_1,x_2;T)\hfill \\ & \ge & P(U_1\left(T\right)<0,U_2\left(T\right)<0)\hfill \\ & \ge & P\left(\sum _{i=1}^{N\left(T\right)}{X}_i^{\left(1\right)}{e}^{-R_1\left({\sigma }_i\right)}-{\delta }_1{p}_1^{*}\left(T\right)-MT>x_1,\sum _{j=1}^{N\left(T\right)}{X}_j^{\left(2\right)}{e}^{-R_2\left({\sigma }_j\right)}-{\delta }_2{p}_2^{*}\left(T\right)-MT>x_2\right)\hfill \\ & \gtrsim & \underset{s,u\ge 0,s+u\le T}{\int \int }(P\left(X_1^{\left(1\right)}{e}^{-R_1(u+s)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\hfill \\ & & +P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2(u+s)}>x_2\right))\lambda \left(du\right)\lambda \left(ds\right)\hfill \\ & & +{\int }_{0-}^{T}P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\lambda \left(du\right).\hfill \end{array}$$

(16)

By the fact that ${\psi }_{sim}(x_1,x_2;T)\le {\psi }_{and}(x_1,x_2;T)$, for all $x_1,x_2>0$ and $T\in \mathsf{\Lambda }$, and the asymptotic result of ${\psi }_{and}(x_1,x_2;T)$, we get that

$${\psi }_{sim}(x_1,x_2;T)\sim {\psi }_{and}(x_1,x_2;T).$$

This completes the proof of Theorem 1. □

Proof of Theorem 2.

By the definition of ${\psi }_{or}(x_1,x_2;T)$, we know that for any $x_1,x_2>0$ and $T\in \mathsf{\Lambda }$,

$$\begin{array}{ccc}& & {\psi }_{or}(x_1,x_2;T)\hfill \\ & =& P\left(\underset{0\le t\le T}{inf}{U}_1\left(t\right)<0\right)+P\left(\underset{0\le t\le T}{inf}{U}_2\left(t\right)<0\right)-{\psi }_{and}(x_1,x_2;T).\hfill \end{array}$$

(17)

From Theorem 2.1 of [12], under the conditions of Theorem 1, for $i=1,2$, we can get for any $T\in \mathsf{\Lambda }$ that

$$\begin{array}{c}\hfill P\left(\underset{0\le t\le T}{inf}{U}_i\left(t\right)<0\right)\sim {\int }_{0-}^T\overline{F_i}\left(x_i{e}^{R_i\left(u\right)}\right)\lambda \left(du\right).\end{array}$$

(18)

Note that, for $i=1,2$, any $x_1,x_2>0$ and $T\in \mathsf{\Lambda }$,

$$\begin{array}{c}\hfill {\int }_{0-}^{T}P\left(X_1^{\left(i\right)}{e}^{-R_i\left(u\right)}>x_i\right)\lambda \left(du\right)\ge P\left(X_1^{\left(i\right)}{e}^{-R_i\left(T\right)}>x_i\right)\lambda \left(T\right).\end{array}$$

(19)

Since $\{R_i\left(t\right);t\ge 0\},i=1,2$, are nonnegative Lévy processes and $F_i\in \mathcal{D},i=1,2$, it follows from Theorem 3.3(iv) of [21] that for any $T\in \mathsf{\Lambda }$ and $i=1,2$,

$$\begin{array}{c}\hfill \overline{F_i}\left(x_i\right)=O\left(P\left(X_1^{\left(i\right)}{e}^{-R_i\left(T\right)}>x_i\right)\right).\end{array}$$

(20)

Thus, by (2), (19) and (20), for any $T\in \mathsf{\Lambda }$, it holds that

$$\begin{array}{ccc}& & \underset{(x_1,x_2)\to (\infty ,\infty )}{lim}{\displaystyle \frac{{\psi }_{and}(x_1,x_2;T)}{{\int }_{0-}^{T}P\left(X_1^{\left(1\right)}{e}^{-R_1\left(u\right)}>x_1\right)\lambda \left(du\right)+{\int }_{0-}^{T}P\left(X_1^{\left(2\right)}{e}^{-R_2\left(u\right)}>x_2\right)\lambda \left(du\right)}}\hfill \\ & \le & \underset{(x_1,x_2)\to (\infty ,\infty )}{lim}{\displaystyle \frac{{\psi }_{and}(x_1,x_2;T)}{P\left(X_1^{\left(1\right)}{e}^{-R_1\left(T\right)}>x_1\right)\lambda \left(T\right)+P\left(X_1^{\left(2\right)}{e}^{-R_2\left(T\right)}>x_2\right)\lambda \left(T\right)}}\hfill \\ & \le & \underset{(x_1,x_2)\to (\infty ,\infty )}{lim}{\displaystyle \frac{2\overline{F_1}\left(x_1\right)\overline{F_2}\left(x_2\right)\lambda \left(T\right)\lambda \left(T\right)+\overline{F_1}\left(x_1\right)\overline{F_2}\left(x_2\right)\lambda \left(T\right)}{P\left(X_1^{\left(1\right)}{e}^{-R_1\left(T\right)}>x_1\right)\lambda \left(T\right)+P\left(X_1^{\left(2\right)}{e}^{-R_2\left(T\right)}>x_2\right)\lambda \left(T\right)}}\hfill \\ & =& 0.\hfill \end{array}$$

(21)

By (17), (18) and (21), we get that (3) holds. This completes the proof of Theorem 2. □

5. Conclusions

In this paper, we investigate the asymptotics of the finite-time ruin probabilities of a dependent bidimensional risk model which contains a stochastic return and Brownian perturbations. The obtained results have improved and extended some related results in the literature. Firstly, this paper considers the risk model with a stochastic return and investigates the case that the price processes of the investment portfolio of the two lines of besiness are two geometic Lévy processes, which improves the situation that the return on investment is a constant interest rate. Secondly, we consider that the claims of each line of business are dependent and obtain the sharp asymptotic bounds for the two kinds of finite-time ruin probabilities ${\psi }_{and}(x_1,x_2;T)$ and ${\psi }_{sim}(x_1,x_2;T)$ for the whole subexponential claims. Finally, the paper also investigates the effect of the perturbations in the risk model on the asymptotics of the finite-time ruin probabilities.