Modern Stochastics: Theory and Applications

In the actuarial literature, a lot of attention is paid to the investigation of the ruin measures such as the ruin probability, the surplus prior to ruin and the deficit at ruin (see, e.g., [6, 34, 39] and references therein). A unified approach to the study of these risk measures together by combining them into one function was proposed by Gerber and Shiu [22], who introduced the expected discounted penalty function for the classical risk model. The so-called Gerber–Shiu function has been investigated further by many authors (see, e.g., [13, 23, 29, 42, 44]) in more general risk models. In those risk models, claim sizes and inter-claim times are assumed to be mutually independent, which simplifies the investigation of the ruin measures. Nevertheless, this assumption has been proved to be very restrictive in some real applications. For instance, in modelling damages caused by natural catastrophic events, the intensity of the catastrophe and the time elapsed since the last catastrophe are expected to be dependent [10, 38]. That is why more and more authors have concentrated on the investigation of risk models with dependence between claim sizes and inter-claim times recently.

Albrecher and Boxma [1] consider a generalization of the classical risk model, where the distribution of the time between two claims depends on the previous claim size (see also [2] for an extension). Albrecher and Teugels [3] apply the random walk approach and allow the inter-claim time and its subsequent claim size to be dependent according to an arbitrary structure. Boudreault, Cossette, Landriault and Marceau [11] consider a particular dependence structure between the inter-claim time and the subsequent claim size and derive the defective renewal equation satisfied by the expected discounted penalty function. In [33], the authors study the ruin probability in a model where the time between two claim occurrences determines the distribution of the next claim size. The ruin probability in a model with independent but not necessarily identically distributed claim sizes and inter-claim times is investigated in [5] (see also references therein).

Cossette, Marceau and Marri [15, 16] deal with an extension of the classical compound Poisson risk model where a dependence structure between the claim size and the inter-claim time is introduced through a Farlie–Gumbel–Morgenstern copula and its generalization. They derive the integro-differential equation and the Laplace transform of the Gerber–Shiu discounted penalty function and concentrate on exponentially distributed claim sizes. Zhang and Yang [47] extend these results to the compound Poisson risk model perturbed by a Brownian motion. In [12, 24, 46], the authors deal with the Sparre Andersen risk model where the inter-claim times follow the Erlang distribution and extend results obtained in [15, 16].

In all these papers, the dependence structure between the claim sizes and the inter-claim times is described by the Farlie–Gumbel–Morgenstern copula. This copula is often used in applications to introduce dependence structures due to its tractability and simplicity. It allows positive and negative dependence as well as independence. Nevertheless, the Farlie–Gumbel–Morgenstern copula has been shown to be somewhat limited since it does not allow the modeling of high dependencies. Indeed, its dependence parameter is $\theta \in [-1,1]$, so its Spearman’s rho and Kendall’s tau are $\rho _{\theta }=\theta /3\in [-1/3,1/3]$ and $\tau _{\theta }=2\theta /9\in [-2/9,2/9]$, respectively (see, e.g., [7, 8, 37] and references therein). This limited range of dependence restricts the usefulness of this copula for modeling. Note that the dependence parameter θ can be easily estimated from real data due to the simple relations between it and the measures of association $\rho _{\theta }$ and $\tau _{\theta }$. For more information on the Farlie–Gumbel–Morgenstern copula, we refer to, e.g., [21, 37] (see also [16] and references therein for applications of this copula). Despite the popularity of the Farlie–Gumbel–Morgenstern copula, other copulas have been used in risk theory, for instance, an Archimedean copula [4], a Gaussian copula [19] and a Spearman copula [25].

Risk models where an insurance company pays dividends to its shareholders are of great interest in risk theory. Dividend strategies for insurance risk models were first proposed by De Finetti [20], who dealt with a binomial model. Barrier strategies for the classical risk model and its different generalizations have been studied in a number of papers (see, e.g., [14, 27, 28, 30, 31, 43, 45]). For optimal dividend problems in insurance risk models, see the monograph by Schmidli [40] and references therein.

Cossette, Marceau and Marri [17, 18] consider the classical risk process with a constant dividend barrier and a dependence structure between claim sizes and inter-claim times introduced through the Farlie–Gumbel–Morgenstern copula. They analyze the Gerber–Shiu function and the expected discounted dividend payments and then concentrate on exponentially distributed claim sizes investigating the impact of the dependence on ruin quantities. The same model is studied in [32], where, in particular, the authors show that the solution to the integro-differential equation for the Gerber–Shiu function is a linear combination of the Gerber–Shiu function with no barrier and the solution to the associated homogeneous integro-differential equation. For some earlier results in this direction, see also [26].

Shi, Liu and Zhang [41] consider the compound Poisson risk model with a threshold dividend strategy and a dependence structure modeled by the Farlie–Gumbel–Morgenstern copula. They derive integro-differential equations for the Gerber–Shiu function and the expected discounted dividend payments paid until ruin as well as renewal equations for these functions, which are used to obtain explicit formulas for them.

The present paper deals with a generalization of the risk model with stochastic premiums introduced and studied in [9] (see also [34]). In contrast to the classical compound Poisson risk model, where premiums arrive with constant intensity and are not random, in this risk model premiums also form a compound Poisson process, i.e. they arrive at random times and their sizes are also random (see also [35, 36] for a generalization of the classical risk model where an insurance company gets additional funds whenever a claim arrives). In [9], claim sizes and inter-claim times are assumed to be mutually independent, and the same assumption is made concerning premium arrivals. In this paper, we suppose that the dependence structures between claim sizes and inter-claim times as well as premium sizes and inter-premium times are modeled by the Farlie–Gumbel–Morgenstern copulas, which allows positive and negative dependence as well as independence. In addition, we suppose that the insurance company pays dividends to its shareholders according to a threshold dividend strategy. To be more precise, this implies that when the surplus is below some fixed threshold, no dividends are paid, and when the surplus exceeds or equals the threshold, dividends are paid continuously at some constant rate. Our subjects of investigation are the Gerber–Shiu function, a special case of which is the ruin probability, and the expected discounted dividend payments until ruin.

The rest of the paper is organized as follows. In Section 2, we describe the risk model we deal with. In Section 3, we derive integral and integro-differential equations for the Gerber–Shiu function. In Section 4, we obtain corresponding equations for the expected discounted dividend payments until ruin. Section 5 deals with exponentially distributed claim and premium sizes in some special cases of the model. Namely, we consider the ruin probability in the model without either dividend payments or dependence, and the expected discounted dividend payments in the model without dependence. In these simpler models, we reduce the integral and integro-differential equations derived in Sections 2 and 3 to linear differential equations and find explicit solutions to these equations. Section 6 provides some numerical illustrations.

Let $(\varOmega ,\mathfrak{F},\mathbb{P})$ be a probability space satisfying the usual conditions, and let all the stochastic objects we use below be defined on it.

In the risk model with stochastic premiums introduced in [9] (see also [34]), claim sizes form a sequence $(Y_{i})_{i\ge 1}$ of non-negative independent and identically distributed (i.i.d.) random variables (r.v.’s) with cumulative distribution function (c.d.f.) $F_{Y}(y)=\mathbb{P}[Y_{i}\le y]$. The number of claims on the time interval $[0,t]$ is a Poisson process $(N_{t})_{t\ge 0}$ with constant intensity $\lambda >0$. Next, premium sizes form a sequence $(\bar{Y}_{i})_{i\ge 1}$ of non-negative i.i.d. r.v.’s with c.d.f. $\bar{F}_{\bar{Y}}(y)=\mathbb{P}[\bar{Y}_{i}\le y]$. The number of premiums on the time interval $[0,t]$ is a Poisson process $(\bar{N}_{t})_{t\ge 0}$ with constant intensity $\bar{\lambda }>0$. Thus, the total claims and premiums on $[0,t]$ equal ${\sum _{i=1}^{N_{t}}}Y_{i}$ and ${\sum _{i=1}^{\bar{N}_{t}}}\bar{Y}_{i}$, respectively. We set ${\sum _{i=1}^{0}}Y_{i}=0$ if $N_{t}=0$, and ${\sum _{i=1}^{0}}\bar{Y}_{i}=0$ if $\bar{N}_{t}=0$. In what follows, we also assume that the r.v.’s $(Y_{i})_{i\ge 1}$ have a probability density function (p.d.f.) $f_{Y}(y)$ and a finite expectation $\mu >0$, and the r.v.’s $(\bar{Y}_{i})_{i\ge 1}$ have a probability density function (p.d.f.) $f_{\bar{Y}}(y)$ and a finite expectation $\bar{\mu }>0$.

We denote a non-negative initial surplus of the insurance company by x. Let $X_{t}(x)$ be its surplus at time t provided that the initial surplus is x. Then the surplus process $(X_{t}(x))_{t\ge 0}$ follows the equation

In [9], the r.v.’s $(Y_{i})_{i\ge 1}$ and $(\bar{Y}_{i})_{i\ge 1}$, and the processes $(N_{t})_{t\ge 0}$ and $(\bar{N}_{t})_{t\ge 0}$ are assumed to be mutually independent. In this paper, we suppose that the claim sizes $(Y_{i})_{i\ge 1}$ and the inter-claim times are not independent but with a dependence structure modeled by a Farlie–Gumbel–Morgenstern copula, and we make the same assumption concerning premium arrivals.

To be more precise, let $(T_{i})_{i\ge 1}$ be a sequence of inter-arrival times of $(N_{t})_{t\ge 0}$. In particular, $T_{1}$ is the time of the first claim. Thus, $(T_{i})_{i\ge 1}$ are i.i.d. r.v.’s with p.d.f. $f_{T}(t)=\lambda {e}^{-\lambda t}$. We assume that $(Y_{i},T_{i})_{i\ge 1}$ are i.i.d. random vectors and for every fixed $i\ge 1$, the dependence structure between $Y_{i}$ and $T_{i}$ is modeled by a Farlie–Gumbel–Morgenstern copula with parameter $\theta \in [-1,1]$, i.e. (see, e.g., [21, 37] for more information on copulas). In other words, a claim size depends on the time elapsed from the previous claim. Therefore, the bivariate c.d.f. of $(Y_{i},T_{i})_{i\ge 1}$ is defined by The corresponding bivariate p.d.f. of $(Y_{i},T_{i})_{i\ge 1}$ is given by where $h_{Y}(y)=f_{Y}(y)(1-2F_{Y}(y))$, $y\ge 0$. Note that the case $\theta =0$ corresponds to the situation where the claim sizes and the inter-claim times are independent.

Next, let $(\bar{T}_{i})_{i\ge 1}$ be a sequence of inter-arrival times of $(\bar{N}_{t})_{t\ge 0}$. In particular, $\bar{T}_{1}$ is the time of the first premium. Therefore, $(\bar{T}_{i})_{i\ge 1}$ are i.i.d. r.v.’s with p.d.f. $f_{\bar{T}}(t)=\bar{\lambda }{e}^{-\bar{\lambda }t}$. We also suppose that $(\bar{Y}_{i},\bar{T}_{i})_{i\ge 1}$ are i.i.d. random vectors and for every fixed $i\ge 1$, the dependence structure between $\bar{Y}_{i}$ and $\bar{T}_{i}$ is modeled by a Farlie–Gumbel–Morgenstern copula with parameter $\bar{\theta }\in [-1,1]$. So the bivariate p.d.f. of $(\bar{Y}_{i},\bar{T}_{i})_{i\ge 1}$ is given by where $h_{\bar{Y}}(y)=f_{\bar{Y}}(y)(1-2F_{\bar{Y}}(y))$, $y\ge 0$. The case $\bar{\theta }=0$ corresponds to the situation where the premium sizes and the inter-premium times are independent. The random vectors $(Y_{i},T_{i})_{i\ge 1}$ and $(\bar{Y}_{i},\bar{T}_{i})_{i\ge 1}$ are assumed to be mutually independent.

From (2) and (3) we obtain the conditional p.d.f.’s of the claim and premium sizes: and

Moreover, we suppose that the insurance company pays dividends to its shareholders according to the following threshold dividend strategy. Let $b>0$ be a threshold. When the surplus is below b, no dividends are paid. When the surplus exceeds or equals b, dividends are paid continuously at a rate $d>0$. Let $({X_{t}^{b}}(x))_{t\ge 0}$ denote the modified surplus process under this threshold dividend strategy. Then where $\mathbb{1}(\cdot )$ is the indicator function.

Let $(D_{t})_{t\ge 0}$ denote the dividend distributing process. For the threshold dividend strategy described above, we have

Next, let $\tau _{b}(x)=\inf \{t\ge 0:{X_{t}^{b}}(x)<0\}$ be the ruin time for the risk process $({X_{t}^{b}}(x))_{t\ge 0}$ defined by (6). In what follows, we omit the dependence on x and write $\tau _{b}$ instead of $\tau _{b}(x)$ when no confusion can arise.

For $\delta _{0}\ge 0$, the Gerber–Shiu function is defined by where $w(\cdot ,\cdot )$ is a bounded non-negative measurable function, ${X_{\tau _{b}-}^{b}}(x)$ is the surplus immediately before ruin and $|{X_{\tau _{b}}^{b}}(x)|$ is a deficit at ruin. Note that if $w(\cdot ,\cdot )\equiv 1$ and $\delta _{0}=0$, then $m(x,b)$ becomes the infinite-horizon ruin probability

For $\delta >0$, the expected discounted dividend payments until ruin are defined by

For simplicity of notation, we also write $m(x)$ and $v(x)$ instead of $m(x,b)$ and $v(x,b)$, respectively, when no confusion can arise. Moreover, we set and Thus, the functions $m_{1}(x)$ and $v_{1}(x)$ are defined on $[0,b]$, the functions $m_{2}(x)$ and $v_{2}(x)$ are defined on $[b,\infty )$ and we have $m_{1}(b)=m_{2}(b)$ and $v_{1}(b)=v_{2}(b)$.

In this section, we deal with exponentially distributed claim and premium sizes, i.e. and

We now present numerical examples for the results obtained in Section 5. The claim and premium sizes are also assumed to be exponentially distributed. Set $\lambda =0.1$, $\bar{\lambda }=2.3$, $\mu =3$ and $\bar{\mu }=0.2$.

Let now the conditions of Theorem 3 hold. Then applying this theorem we can calculate the ruin probability for $x\in [0,\infty )$ in the model without dividend payments for different values of θ:

Table 1 presents the results of calculations for some values of x.

Next, we denote by $\psi _{0}(x)$ the ruin probability in this model where $\theta =0$. It is given by (see [9, 34]). In our example, $\psi _{0}(x)\approx 0.666667{e}^{-0.111111x}$. The values of $\psi _{0}(x)$ for some x are given in Table 2.

Let now the conditions of Theorems 4 and 5 hold. Set additionally $b=5$, $d=0.1$ and $\delta =0.01$. Applying Theorems 4 and 5 we can calculate the ruin probability $\psi (x)$ and the expected discounted dividend payments until ruin $v(x)$:

The results of calculations for some values of x are given in Table 2.

The results presented in Tables 1 and 2 show that the positive dependence between the claim sizes and the inter-claim times decreases the ruin probability and the negative dependence increases it. This conclusion seems to be natural. Indeed, in the case of negative dependence, the situation where large claims arrive in short time intervals is more probable, which obviously leads to ruin in the near future. Moreover, it is easily seen from Table 2 that dividend payments substantially increase the ruin probability, which is also an expected conclusion.