Precise large deviations for aggregate claims in a two-dimensional compound dependent risk model

1.   Introduction

This paper will investigate a two-dimensional compound risk model. In this risk model, an insurance company has two dependent classes of business sharing a common claim-number process, which is a compound renewal counting process. Let the inter-arrival times of events $\{\theta_{k}, k\geq 1\}$ be a sequence of independent and identically distributed (i.i.d.) nonnegative random variables (r.v.s) with finite mean $\beta^{-1} > 0$. Let $Z_k$ be the number of claims caused by the $k$th $(k\geq 1)$ event. Suppose that $\{Z_{k}, k\geq 1\}$ are i.i.d. positive integer r.v.s with finite mean $\mu_{Z}$ and independent of $\{\theta_{k}, k\geq 1\}$. Then the number of events up to time $t\geq 0$ is denoted by

and the number of claims up to time $t\geq0$ is denoted by

which is a compound renewal counting process. Set $\theta (t) = E(N(t))$ and $\lambda(t) = E(\Lambda(t))$, $t\geq 0$, then $\theta(t)/t\rightarrow \beta$ as $t\rightarrow \infty$ and $\lambda(t) = \mu_Z\theta(t)$, $t\geq 0$. The claim-amount vectors $\vec{X}_{k} = \left(X_{1 k}, X_{2 k}\right)^{T}, k\geq 1$ are i.i.d. copies of $\vec{X} = \left(X_{1 }, X_{2 }\right)^{T}$ with finite mean vector $\vec{\mu} = \left(\mu_{1 }, \mu_{2 }\right)^{T}$. Assume that $X_1$ and $X_2$ are nonnegative r.v.s with distributions $F_1$ and $F_2$, respectively. Their joint distribution is denoted by $F_{12}(x_1, x_2) = P(X_1\leq x_1, X_2 \leq x_2)$ and their joint survival function is $\overline{F_{12}}(x_1, x_2) = P(X_1 > x_1, X_2 > x_2)$. Then the aggregate amount of claims up to time $t\geq 0$ is expressed as

This paper will investigate the precise large deviations of $\vec{S}(t), t\geq 0.$

In this paper, we assume that $\{Z_k, k\geq 1\}$ are independent of $\{\vec{X}_{k}, k\geq 1\}$ and $\{(\vec{X}_{k}, \theta_{k}), k\geq 1\}$ are i.i.d. random vectors with generic pair $(\vec{X}, \theta).$ This paper mainly considers for each $k\geq 1$, $X_{1k}$, $X_{2k}$ and $\theta_{k}$ may be dependent and the claims have heavy-tailed distributions. In the following section some heavy-tailed distribution classes will be given.

Without special statement, in this paper a limit is taken as $t\to\infty$. For a real-valued number $a$, let $a^+ = \max\{0, a\}$ and $a^- = -\min\{0, a\}$. Denote $[a]$ by the large integer that does not exceed $a$. {For two vectors $\vec{y} = (y_1, y_2)^T$ and $\vec{z} = (z_1, z_2)^T$, $\vec{y} > \vec{z}$ (or $\geq$) means $y_{i} > z_{i}$ (or $\geq$), $i = 1, 2$. } For two nonnegative functions $a(\cdot)$ and $b(\cdot)$, we write $a(t)\lesssim b(t)$ if $\limsup a(t)/b(t)\leq 1$, write $a(t)\gtrsim b(t)$ if $\liminf a(t)/b(t)\geq 1$, write $a(t)\sim b(t)$ if $\lim a(t)/b(t) = 1$, and write $a(t) = o(b(t))$ if $\lim a(t)/b(t) \; = 0$. For two positive bivariate functions $g(\cdot, \cdot)$ and $h(\cdot, \cdot)$, we write $g(x, t)\lesssim h(x, t)$, as $t\to\infty$, holds uniformly for $x\in \Delta \not = \phi$, if

We write $g(x, t)\gtrsim h(x, t)$, as $t\to\infty$, holds uniformly for $x\in \Delta \not = \phi$, if

In the following, we give some heavy-tailed distribution classes. For a proper distribution $V$ on $(-\infty, \infty)$, let $\overline{V} = 1-V$ be the tail of $V$. Say that a distribution $V$ on $(-\infty, \infty)$ is heavy-tailed, if for any $s > 0$,

Otherwise, say that $V$ is light-tailed. The dominated variation distribution class $\mathscr{D}$ is an important class of heavy-tailed distributions. Say that a distribution $V$ on $(-\infty, \infty)$ belongs to the class $\mathscr{D}$, if for any $y\in(0, 1)$,

The slightly smaller class is the class $\mathscr{C}$, which consists of all distributions with consistently varying tails. Say that a distribution $V$ on $(-\infty, \infty)$ belongs to the class $\mathscr{C}$ if

Another class is the long-tailed distribution class $\mathscr{L}$. Say that a distribution $V$ on $(-\infty, \infty)$ belongs to the class $\mathscr{L}$ if for any $y > 0$,

It is well known that these distribution classes have the following relationships:

(see, e.g., Cline and Samorodnitsky ^[5], Embrechts et al. ^[7]).

For a distribution $V$ on $(-\infty, \infty)$, let

We call $J_{V}^+$ the upper Matuszewska index of $V$. For the details of the Matuszewska index one can see Bingham et al. ^[2].

In recent years, more and more researchers pay attention to multi-dimensional risk models and study the precise large deviations of aggregate amount of claims, see e.g. Wang and Wang ^[19], Wang and Wang ^[20], Lu ^[12], Tian and Shen ^[14] and so on. Recently, Fu et al. ^[8] studied the precise large deviations of $S_{N(t)} = \sum_{k = 1}^{N(t)}X_k$, $t\geq 0$ under the following assumptions.

Assumption 1.1. The random vector $\left(X_{1}, X_{2}\right)$ has a survival copula $\hat{C}(\cdot, \cdot)$ satisfying

where $g_{u}(\cdot)$ is a finite positive function.

Definition 2.2.2 of Nelsen ^[13] gave the definition of copula. A copula is a function $C$ from $[0, 1]\times [0, 1]\to [0, 1]$ with the following properties:

(1) For every $u, v\in [0, 1]$, $C(u, 0) = C(0, v) = 0$, $C(u, 1) = u$ and $C(1, v) = v$.

(2) For every $u_1, u_2, v_1, v_2\in [0, 1]$ such that $u_1\leq u_2$ and $v_1\leq v_2$,

The Sklar's theorem (i.e. Theorem 2.3.3 of Nelsen ^[13]) states that for the r.v.s $X_1$ and $X_2$ in Assumption 1.1, there exists a copula $C$ such that for all $x_i\in (-\infty, \infty), i = 1, 2$,

Let $\hat{C}(u, v) = u+v-1+C(1-u, 1-v), u, v\in [0, 1]$, then for all $x_i\in (-\infty, \infty), i = 1, 2$,

We call $\hat{C}$ as the survival copula of $X_1$ and $X_2$ (see (2.6.1) and (2.6.2) of Nelsen ^[13]).

Assumption 1.2. There exists a nonnegative random variable $\theta^{*}$ with finite mean such that $\theta$ conditional on $\left(X_{i} > x_{i}\right), i = 1, 2$, is stochastically bounded by $\theta^{*}$ for all large $x_{1}$ and $x_{2};$ i.e., there exists some $\vec{x}_{0} = \left(x_{10}, x_{20}\right)^{T}$ such that it holds for all $\vec{x} = \left(x_{1}, x_{2}\right)^{T} > \vec{x}_{0}$ and $t \in[0, \infty)$ that

This paper still uses the above two assumptions. We will investigate the precise large deviations of the aggregate amount of claims in a two-dimensional compound risk model. For the one-dimensional compound risk model, there are many papers studying the aggregate amount of claims, such as Tang et al. ^[15], Aleškevičienė et al. ^[1], Konstantinides and Loukissas ^[11], Yang et al. ^[22], Guo et al. ^[9], Wang and Chen ^[18], Yang et al. ^[23], Wang et al. ^[17], Xun et al. ^[21] and so on. For a two-dimensional compound risk model researchers mainly studied the ruin probabilities, such as Cai and Li ^[4], Delsing et al. ^[6] and so on. This paper will consider the precise large deviations of compound sum (1.1) in a two-dimensional compound risk model. The following is the main result of this paper.

Theorem 1.1. Consider the model (1.1). Suppose that Assumptions 1.1 and 1.2 are satisfied, $F_{i} \in \mathscr{C}$, $i = 1, 2$ and there exists a constant $\alpha_{Z} > 2\max\{J_{F_1}^{+}, J_{F_2}^{+}\}+4$ such that $EZ_1^{\alpha_{Z}} < \infty$. Then for any $\vec{\gamma} = \left(\gamma_{1}, \gamma_{2}\right)^{T} > \vec{0}$,

holds uniformly for all $\vec{x} \geq \vec{\gamma} \lambda(t)$.

Remark 1.1. In the two-dimensional compound renewal risk model (1.1), if $Z_k\equiv1, k\geq 1$, then model (1.1) degenerates into the classic two-dimensional renewal risk model. In the classic two-dimensional renewal risk model, suppose that $F_i\in \mathscr{C}$, i = 1, 2 and Assumptions 1.1 and 1.2 are satisfied. Then from Theorem 1.1 the main result of Fu et al. ^[8] can be obtained.

The proof of Theorem 1.1 will be given in the following section.

2.   Proof of the main result

By Assumption 1.2, we introduce two independent nonnegative r.v.s $\theta_{1} ^{**}$ and $\theta_{2}^{**}$, which have the same distributions as $\theta$ conditional on $\{X_1 > x_1\}$ and $\{X_2 > x_2\}$, respectively. Assume that $\theta_{1}^{**}$ and $\theta_{2}^{**}$ are independent of all other r.v.s. Let $\tau_{1}^{* *} = \theta_{1}^{* *}, \tau_{2}^{* *} = \theta_{1}^{* *}+\theta_{2}^{* *}, \tau_{n}^{* *} = \; \theta_{1}^{* *}+\theta_{2}^{* *}+\sum_{i = 3}^{n} \theta_{i}$, $n\geq 3$, and define

Set $\Lambda^{**}(t) = \sum_{k = 1}^{N^{**}(t)}Z_k, \; t\geq 0.$ The following relation implies that for each $t\geq 0$, $\Lambda^{* *}(t)$ is also identically distributed as $\Lambda(t)$ conditional on $\{X_1 > x_1, X_2 > x_2\}$. In fact, noticing the independence assumption between $\{Z_k, k\geq 1\}$ and $(\vec{X}, \theta)$, it holds for $t\geq 0$, $n\geq1$ and $x_1, x_2\geq0$ that

Before giving the proof of Theorem 1.1, we first give some lemmas. The first lemma gives a property about $\Lambda^{**}(t), t\geq 0$.

Lemma 2.1. In addition to Assumption 1.2, assume that $\operatorname{Var} \theta < \infty$. Then it holds for every $0 < \delta < \beta$ and every functions $a(t)$ and $b(t)$ that

where $a(\cdot):[0, \infty)\rightarrow (0, \infty)$ with $a(t)\uparrow \infty$ and $b(\cdot):[0, \infty)\rightarrow (0, \infty)$ with $b(t)\uparrow \infty$.

Proof. Using the same method of the proof of Lemma 3.4 of Bi and Zhang ^[3], we can get that

In the following we will prove for any $\epsilon > 0$

For the above $\epsilon > 0$, by (2.3) and the law of large number for i.i.d r.v.s, it holds uniformly for $x_{1} \geq a(t)$ and $x_{2} \geq b(t)$ that

and

In the following, we prove (2.2). Since $\lambda(t)\sim \mu_{Z}\beta t$, {it holds for any $0 < \epsilon < \delta(\mu_{Z} \beta)^{-1}$ that} $(1-\epsilon)\mu_{Z}\beta t\leq \lambda(t)\leq (1+\epsilon)\mu_{Z}\beta t$. Thus by (2.3) and (2.4), it holds uniformly for $x_1 \geq a(t)$ and $x_2 \geq b(t)$ that

Similarly, it holds uniformly for $x_1 \geq a(t)$ and $x_2 \geq b(t)$ that

which combining with (2.6) yields that (2.2) holds.

The following lemma is Lemma 3.2 of Fu et al. ^[8].

Lemma 2.2. Let $\left\{\vec{X}_{k}, k\geq 1\right\}$ be a sequence of $i. i. d$. random vectors with finite mean vector $\vec{\mu}$. In addition to Assumptions 1.1 and 1.2, suppose that $F_{i} \in \mathscr{C}$, $i = 1, 2$. Then for any $\vec{\gamma} = (\gamma_{1}, \gamma_{2})^T > \overrightarrow{0}$, it holds uniformly for all $\vec{x} = (x_1, x_2)^T \geq \vec{\gamma} n$ that

as $n \rightarrow \infty$, where $\vec{S}_{n} = (S_{1n}, S_{2n})^T = \sum_{k = 1}^{n} \vec{X}_{k}$.

From Proposition 2.2.1 of Bingham et al. ^[2], we obtain

Lemma 2.3. If $V \in \mathscr{D}$ then for every $p > J_{V}^{+}$, there are positive constants C and $x_0$ such that

holds for all $xy \geq x \geq x_0$.

The next lemma comes from Lemma 1(i) of Kočetova et al. ^[10].

Lemma 2.4. Let the inter-arrival times $\left\{\theta_{k}, k \geq 1 \right\}$ form a sequence of i.i.d. nonnegative r.v.s with common mean $\beta^{-1} \in(0, \infty)$. Then it holds for every $a > \beta$ and some $b > 1$ that

The last lemma is a restatement of Lemma 2.3 of Tang ^[16].

Lemma 2.5. Let $\{\xi_{k}, k\geq 1 \}$ be i.i.d. real-valued r.v.s with common distribution $V$ and mean 0 satisfying $E\left(\xi_{1}^{+}\right)^{r} < \infty$ for some $r > 1$. Then for each fixed $\gamma > 0$ and $p > 0$, there exist positive numbers $v$ and $C = C(v, \gamma)$ irrespective to $x$ and $n$ such that for all $x \geq \gamma n$ and $n\geq 1$

Proof of Theorem 1.1: Without special statement, in this proof a limit relation is understood as valid uniformly for all $\vec{x} \geq \vec{\gamma} \lambda(t)$ as $t \rightarrow \infty$. We will show the following two relations

and

We first prove (2.8). For any $0 < \delta < 1$, it holds that for $x_i > 0, i = 1, 2$ and $t > 0$

For $I_{1}$, by Lemma 2.2 it holds that

where in the third step Lemma 2.2 is used, which is due to the fact that for small $\delta$ such that $\gamma_i- \mu_{i} \delta > 0$, and for any $0 < \gamma_{i}^{'} < \frac{\gamma_{i}-\mu_{i}\delta}{1+\delta},$ it holds that $x_{i}+\mu_{i} \lambda (t)-\mu_{i}[(1+\delta) \lambda (t)] \geq \gamma_{i}^{'} [(1+\delta) \lambda (t)]$ for $x_{i} \geq \gamma_{i} \lambda (t), i = 1, 2$. By $F_{i} \in \mathscr{C}, i = 1, 2$, we have

For $I_2$, take any $0 < \varepsilon < \frac{\delta \mu_{Z} \beta}{\delta+\beta+1}$ we have

We first estimate $K_1$. Letting $p > \max\{J_{F_1}^{+}, J_{F_2}^{+}\},$ it follows from Assumption 1.1 and Lemma 2.3 that there exists some positive constant $C$ such that for any $0 < \varepsilon < \mu_{Z} \beta$

In the following, interchanging the order of sums yields that

Since $\lambda(t)\sim\mu_{Z}\beta t,$ for sufficiently large $t$,

Since $\frac{(1+\delta)(\mu_{Z}\beta-\varepsilon)}{\varepsilon +\mu_{Z}} > \beta$, by (2.16) and Lemma 2.4 it holds that

We continue to deal with $K_2$. As $K_1$, by Assumption 1.1 there exists positive constant $C$ such that

By Lemma 2.5, for fixed $\tilde{\gamma} > 0$ and $\tilde{p} > 0$ there exist some positive $v$ and $C_1$ such that

where by taking $\tilde{\gamma} = \varepsilon$ and $\tilde{p} > 2p+3$, Markov's inequality and (2.19) it holds that

where the last step is due to $\alpha_{Z}-2p-3 > 1$ and $\tilde{p}-2p-2 > 1$.

By (2.13), (2.17) and (2.20) it holds that

By (2.12) and (2.21), we get (2.8) holds.

In the following we prove (2.9). For small enough $0 < \delta < 1$ and $\nu > 1$,

To estimate $P_1$. Similarly to (2.1), we can check that $N^{**}(t)$ is also identically distributed as $N(t)$ conditional on $\{X_{1i} > x_1, X_{2j} > x_2\}$. Following the similar method of (3.7) in Fu et al. ^[8] only by replacing event $\{N(t) = n\}$ with event $\{\Lambda(t) = n\}$, together with Lemma 2.1, we can get

Hence, for $F_i \in \mathscr{C}, i = 1, 2,$ we have

As for $P_2$ and $P_3$, following the similar argument as Fu et al. (2021) we can get

and

By (2.22)–(2.25) we get (2.9) holds.

3.   Conclusions

This paper studies a dependent two-dimensional compound risk model with heavy-tailed claims. We mainly investigate the case that there exists a size-dependent structure between the claim sizes and inter-arrival times. Using the probability limiting theory we give the precise large deviations for aggregate amount of claims in the compound risk model.

Acknowledgments

This work is supported by the 333 High Level Talent Training Project of Jiangsu Province and the Jiangsu Province Key Discipline in the 14th Five-Year Plan.

Conflict of interest

The authors declare no conflicts of interest.