On the expected discounted penalty functions for two classes of risk processes

doi:10.1016/j.insmatheco.2004.12.002

Insurance: Mathematics and Economics

Volume 36, Issue 2, 22 April 2005, Pages 179-193

https://doi.org/10.1016/j.insmatheco.2004.12.002 Get rights and content

Abstract

In this paper, we consider the expected discounted penalty (Gerber–Shiu) functions for a risk model involving two independent classes of insurance risks. We assume that the two claim number processes are independent Poisson and generalized Erlang(2) processes, respectively. Laplace transforms of two types of the Gerber–Shiu functions at ruin are derived from an integro-differential equations system. Explicit results are derived when the claims from both classes are exponentially distributed. Finally, asymptotic results are obtained when the compound Poisson process converges weakly to a Wiener process. Numerical illustrations are also given.

Introduction

Consider an insurance surplus process $U (t) = u + c t - S (t), t \geq 0,$ where u is the amount of initial surplus, c the constant rate of premium, ${S (t); t \geq 0}$ the aggregate claim amount process. In this paper, we assume that $S (t)$ is generated from two classes of insurance risks, i.e., $S (t) = S_{1} (t) + S_{2} (t) = \sum_{i = 1}^{N_{1} (t)} X_{i} + \sum_{i = 1}^{N_{2} (t)} Y_{i}, t \geq 0,$ where ${X_{i}}_{i \geq 1}$ are the claim sizes form the first class, assumed to be i.i.d. positive random variables with a common distribution function P and density p, while ${Y_{i}}_{i \geq 1}$ are the claim sizes from the second class, assumed to be i.i.d. positive random variables with a common distribution function Q and density q. Denote by $μ_{X}$ and $μ_{Y}$ the means of X and Y, and by $\hat{p} (s) = \int_{0}^{\infty} e^{- s x} p (x) d x$ and $\hat{q} (s) = \int_{0}^{\infty} e^{- s x} q (x) d x$ the Laplace transforms of p and q, respectively.

The claim number process ${N_{1} (t); t \geq 0}$ is assumed to be a Poisson process with parameter $λ$ , that is, the corresponding claim inter-arrival times, denoted by ${W_{i}}_{i \geq 1}$ , are i.i.d. exponentially distributed random variables with parameter $λ$ . By contrast, ${N_{2} (t); t \geq 0}$ is a renewal process with i.i.d. claim inter-arrival times ${V_{i}}_{i \geq 1}$ , which are independent of ${W_{i}}_{i \geq 1}$ and generalized Erlang(2) distributed, i.e., $V_{i} ≔ L_{i 1} + L_{i 2}$ , where the ${L_{i 1}}_{i \geq 1}$ are i.i.d. exponentially distributed random variables with parameter $λ_{1}$ and ${L_{i 2}}_{i \geq 1}$ are i.i.d. exponentially distributed random variables with parameter $λ_{2}$ (possibly different from $λ_{1}$ ).

We finally assume that ${X_{i}}_{i \geq 1}$ and ${Y_{i}}_{i \geq 1}$ are mutually independent, also independent of $N_{1} (t)$ and $N_{2} (t)$ , and $c > λ μ_{X} + [λ_{1} λ_{2} / (λ_{1} + λ_{2})] μ_{Y}$ , providing a positive loading condition.

Now define $T = \inf {t \geq 0 : U (t) < 0} (\infty, otherwise)$ to be the ruin time, and $Ψ (u) = P (T < \infty | U (0) = u), u \geq 0,$ to be the ultimate ruin probability. Further define J to be the cause-of-ruin random variable; $J = j$ , if the ruin is caused by a claim of class j, $j = 1$ , 2, then ruin probability $Ψ (u)$ can be decomposed as $Ψ (u) = Ψ_{1} (u) + Ψ_{2} (u)$ , where $Ψ_{j} (u) = P (T < \infty, J = j | U (0) = u), u \geq 0, j = 1, 2,$ is the ruin probability due to a claim of class j.

Let $w_{j} (x, y),$ for $x, y \geq 0, j = 1, 2$ , be the non-negative values of two possibly distinct penalty functions. For $δ > 0$ , and $j = 1, 2$ , define $ϕ_{j} (u) = E [e^{- δ T} w_{j} (U (T -), | U (T) |) I (T < \infty, J = j) | U (0) = u], u \geq 0,$ to be the expected discounted penalty (Gerber–Shiu) function at ruin, if the ruin is caused by a claim of class j, for the surplus $U (T -)$ before ruin and the deficit $| U (T) |$ at ruin.

In the classical risk model, due to the strong Markov property of the surplus process, the Gerber–Shiu function is time homogenous, i.e., it is independent of the time at which the surplus process is observed. However, for our risk model, the Gerber–Shiu functions are no longer time homogeneous, due to the assumption that the claim inter-arrival times from the second class are Erlang(2) distributed. Therefore, for the Gerber–Shiu functions, defined in (2), we assume that a claim from the second class occurs exactly at time 0. More generally, we can define the Gerber–Shiu functions, denoted by $ϕ_{j} (u, τ)$ , as bivariate functions of current reserve u and the length of time $τ$ , elapsed since the time of the last claim from the second class (the surplus process renews itself at these points). The quantities we are interested in are $ϕ_{j} (u, 0) = ϕ_{j} (u), j = 1$ , 2, and $ξ_{j} (u) = E [e^{- δ (T - t)} w_{j} (U (T -), | U (T) |) I (T < \infty, J = j) | L_{11} = t, U (t) = u],$ the Gerber–Shiu functions at the time of the realization of ${L_{i 1}}_{i \geq 1}$ . Then by the law of total probability, for $j = 1$ , 2, we have $ϕ_{j} (u, τ) = ϕ_{j} (u) P (L_{11} > τ) + ξ_{j} (u) P (L_{11} < τ) = e^{- λ_{1} τ} ϕ_{j} (u) + (1 - e^{- λ_{1} τ}) ξ_{j} (u) .$

The Erlang distribution is one of the most commonly used distributions in queuing theory, which is closely related to risk theory; e.g., Asmussen, 1987, Asmussen, 1989. Recently, some papers have discussed how methods and results for the classical risk model can be adapted to a Sparre Andersen model in which claims occur following an Erlang or a generalized Erlang process. See Dickson, 1998, Dickson and Hipp, 1998, Dickson and Hipp, 2001, Cheng and Tang, 2003, Li, 2003, Lin, 2003, Gerber and Shiu, 2003a, Gerber and Shiu, 2003b, Gerber and Shiu, 2005, Dickson and Drekic, 2004, Li and Garrido, 2004, Sun and Yang, 2004, and references therein.

Yuen et al. (2002) consider the non-ruin probability for a risk process involving two dependent classes of insurance business, which can be transformed to a surplus process with two independent classes of business for which one claim number process in a Poisson and the other is a renewal process with Erlang(2) claim inter-arrival times. Explicit results are given only for exponentially distributed claim amounts. In this paper, we consider a risk process with two classes of risks, one is a compound Poisson process, the other is a compound renewal process with claim inter-arrival times being generalized Erlang(2) distributed, using the celebrated Gerber–Shiu expected discounted penalty function. The rest of the paper is organized as follows. In Section 2, we derive a system of integro-differential equations for $ϕ_{1} (u)$ and $ϕ_{2} (u)$ . Section 3 discusses a generalized Lundberg’s fundamental equation and its roots. Laplace transforms of functions $ϕ_{1} (u)$ and $ϕ_{2} (u)$ , as well as $ϕ_{1} (0)$ and $ϕ_{2} (0)$ , are obtained in Section 4. Explicit results when claim sizes are exponentially distributed are obtained in Section 5. Finally, in Section 6, Laplace transforms of the Gerber–Shiu functions (caused by a claim and due to oscillations), in a Sparre Andersen risk model perturbed by diffusion, are derived by letting the compound Poisson process converge weakly to a Brownian motion.

Section snippets

System of integro-differential equations

Let $M = W_{1} \land L_{11}$ , then for $u \geq 0$ , $ϕ_{1} (u) = \int_{0}^{\infty} P (M = t, M = L_{11}) e^{- δ t} ξ_{1} (u + c t) d t + \int_{0}^{\infty} P (M = t, M = W_{1}) e^{- δ t} \times [\int_{0}^{u + c t} ϕ_{1} (u + c t - x) p (x) d x + \int_{u + c t}^{\infty} w_{1} (u + c t, x - u - c t) p (x) d x] d t .$ Since $P (M = W_{1}) = λ / (λ + λ_{1})$ , $P (M = L_{11}) = λ_{1} / (λ + λ_{1})$ , and $P (M > t | M = W_{1}) = P (M > t | M = L_{11}) = e^{- (λ + λ_{1}) t}$ , then (3) can be rewritten as $ϕ_{1} (u) = λ_{1} \int_{0}^{\infty} e^{- (λ + λ_{1} + δ) t} ξ_{1} (u + c t) d t + λ \int_{0}^{\infty} e^{- (λ + λ_{1} + δ) t} [\int_{0}^{u + c t} ϕ_{1} (u + c t - x) p (x) d x + \int_{u + c t}^{\infty} w_{1} (u + c t, x - u - c t) p (x) d x] d t .$

Let $Z = W_{1} \land L_{12}$ , then for $u \geq 0$ , $ξ_{1} (u) = \int_{0}^{\infty} P (Z = t, Z = L_{12}) \int_{0}^{u + c t} ϕ_{1} (u + c t - x) q (x) d x d t + \int_{0}^{\infty} P (Z = t, M = W_{1}) \times [\int_{0}^{u + c t} ξ_{1} (u + c t - x) p (x) d x + \int_{u + c t}^{\infty} w_{1} (u + c t, x - u - c t) p (x) d x] d t .$ Similarly,

Generalized Lundberg fundamental equation

Let $T_{0} = 0, T_{k} = \sum_{j = 1}^{k} V_{j}$ be the arrival time of the k-th claim from the second class, thus $U_{0} = 0$ , and for $k = 1, 2, \dots$ , $U_{k} = U (T_{k}) = u + c T_{k} - \sum_{j = 1}^{k} Y_{j} - \sum_{i = 1}^{N (T_{k})} X_{i} \overset{d}{=} u + \sum_{j = 1}^{k} [c V_{j} - Y_{j} - \sum_{i = 1}^{N (V_{j})} X_{i j}],$ is the surplus immediately after the k-th claim from the second class, where $\overset{d}{=}$ means equality in distribution, and ${X_{i j}}_{i, j \geq 1}$ is a sequence of i.i.d. random variables with the same distribution as that of $X_{1}$ . We seek a number $s \in C$ such that the process ${e^{- δ T_{k} + s U_{k}}}_{k \geq 0}$ forms a martingale. Here, the martingale condition is $E [e^{(c s - δ)}]$

Laplace transforms

Henceforth, we focus our interest on the functions $ϕ_{j} (u)$ , $j = 1$ ,2. Their Laplace transforms can be derived as follows.

As in Dickson and Hipp (2001), we define an operator $T_{r}$ of a real-valued function f, with respect to a complex number r, to be $T_{r} f (x) = \int_{x}^{\infty} e^{- r (y - x)} f (y) d y, x \geq 0 .$ It is clear that the Laplace transform of f, $\hat{f} (s)$ , can be expressed as $T_{s} f (0)$ , and that for distinct $r_{1}$ and $r_{2}$ , $T_{r_{1}} T_{r_{2}} f (x) = T_{r_{2}} T_{r_{1}} f (x) = \frac{T_{r_{1}} f (x) - T_{r_{2}} f (x)}{r_{2} - r_{1}}, x \geq 0 .$ Further properties of the operator $T_{r}$ can be found in Li and Garrido

Explicit results for exponential claim size distributions

We now consider the cases where both the claim size distributions p and q are exponentially distributed, with Laplace transforms $\hat{p} (s) = α / (s + α)$ and $\hat{q} (s) = β / (s + β)$ , respectively.

It turns out that Eqs. (17), (22) can be transformed to expressions by multiplying both denominators and numerators by ${(s + α)}^{2} (s + β)$ : ${\hat{ϕ}}_{1} (s) = \frac{(s - ρ_{1}) (s - ρ_{2}) {(s + α)}^{2} (s + β) m_{1} (s)}{λ_{1} λ_{2} {(s + α)}^{2} (s + β) [γ_{δ} (s) - \hat{q} (s)]}, s \in C,$ ${\hat{ϕ}}_{2} (s) = \frac{(s - ρ_{1}) (s - ρ_{2}) {(s + α)}^{2} (s + β) m_{2} (s)}{λ_{1} λ_{2} {(s + α)}^{2} (s + β) [γ_{δ} (s) - \hat{q} (s)]}, s \in C,$ where in the numerators of (24), (25) ${(s + α)}^{2} (s + β) m_{1} (s) = λ (s +$

Diffusion approximation

In this section, we consider the limit case when the compound Poisson process converges weakly to a Wiener process. We hereby consider the following family of surplus processes: $U_{ɛ} (t) = u + (c + c_{ɛ}) t - ɛ N_{1} (t) - \sum_{i = 1}^{N_{2} (t)} Y_{i}, ɛ \in R^{+},$ in which all the assumptions are the same as that for model (1), except for the premium rate that is $c + c_{ɛ}$ , the counting process $N_{1}$ is a Poisson process with parameter $λ_{ɛ}$ , and the claims from the first class are constant of size $ɛ$ , while the corresponding positive loading condition

Concluding remarks

In this paper, we show how to apply the expected discounted penalty functions to a risk process with two independent classes of insurance risks, one is from the classical risk process, the other is from a generalized Erlang(2) risk process. Explicit results are derived when the claims from both classes are exponentially distributed. The decomposition of the ruin probability into ruin probabilities, caused by claims from class one and class two, is given as one of the illustrations.

The results

Acknowledgments

The authors would like to thank Prof. José Garrido and an anonymous referee for their valuable comments and suggestions which lead to the significant improvement of the paper. This research was funded by a Post-Graduate Scholarship of the Natural Sciences and Engineering Research Council of Canada (NSERC).

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On the expected discounted penalty functions for two classes of risk processes

Abstract

Introduction

Section snippets

System of integro-differential equations

Generalized Lundberg fundamental equation

Laplace transforms

Explicit results for exponential claim size distributions

Diffusion approximation

Concluding remarks

Acknowledgments

Insurance: Math. Econ.

Insurance: Math. Econ.

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Applied Probability and Queues