On a perturbed compound Poisson model with varying premium rates

In this paper, we consider a perturbed compound Poisson model with varying premium rates. The surplus process is observed at a sequence of review times. The effective premium rate is adjusted according to the surplus increment between the inter-review times. We study the Gerber-Shiu functions by Laplace transform method. When the claim size density is a combination of exponentials, the explicit expressions for the Laplace transforms of ruin time are derived.


1.
Introduction. In insurance risk theory, the compound Poisson model perturbed by diffusion is defined by where u ≥ 0 is the initial surplus, and c > 0 is the constant premium rate. The aggregate claims process S t = Nt j=1 X j is a compound Poisson process, where {N t } t≥0 is a homogeneous Poisson process with intensity λ > 0, and {X j } j≥1

ZHIMIN ZHANG, YANG YANG AND CHAOLIN LIU
is a sequence of i.i.d. r.v.'s with common density f X (on (0, ∞)), mean µ X = ∞ 0 xf X (x)dx and Laplace transformf X (s) = ∞ 0 e −sx f X (x)dx. Finally, {B t } t≥0 is a standard Brownian motion starting from zero, and σ > 0 is a diffusion parameter.
The perturbed compound Poisson model was first proposed to extend the classical risk model [5]. Since then, a lot of contributions to model (1) have been made in the literature. See e.g. [6,13,14,17,18,4,10], to name a few. We note that it is typically assumed that the premium rate is constant over time. However, the constant premium rate assumption is often justified at the macro level by assuming that the insurance company's portfolio is stable over time. Although the diffusion component can describe uncertainty of the premium rate, it can not effectively represent the point view of the company. [19] considered a perturbed risk model, where the premium rates depend on the claim sizes. In their paper, the premium rate may be adjusted at every claim arrival time. [9] studied a risk model with varying premium rates, where the effective premium rate is determined by the surplus increments between successive review times. In [9], it is assumed that the inter-review time has a combination-of-exponentials density. Although the combination-of-exponentials density can approximate any continuous density function on (0, ∞), the constant inter-review time assumption is more appropriate from the point view of practical applications.
In this paper, we modify model (1) by allowing the premium rate to vary according to the recent claim experience. We consider the embedded premium policy proposed by Li et al. [9]. Let 0 = Z 0 < Z 1 < Z 2 < · · · be a sequence of successive review times, and for j = 1, 2, · · · , define η j (η j = c 1 or c 2 ) to be the effective premium rate between review times Z j−1 and Z j , where we assume that 0 < c 1 < c 2 < ∞. For j = 1, 2, · · · , let T j = Z j − Z j−1 be the jth inter-review time, and conditioning on {η j } j≥1 , it is assumed that the inter-review times {T j } j≥1 are i.i.d, as well as independent of other random quantities. The premium rate is adjusted as follows: (1) if the surplus increment between Z j−1 and Z j is larger than zero, the effective premium rate η j+1 in the next inter-review time interval (Z j , Z j+1 ) is c 1 ; (2) if the surplus increment between Z j−1 and Z j is smaller than zero, the effective premium rate η j+1 in the next inter-review time interval (Z j , Z j+1 ) is c 2 . Furthermore, we assume that c 1 > λµ X so that the safety loading condition holds. After the above modification, we denote the surplus process by {U t } t≥0 .
In this paper, we assume that the inter-review times are Erlang (γ, n) distributed with density where γ > 0 is the scale parameter, and n is the shape parameter that is a positive integer. The Erlang distribution has been widely used in risk theory. On the one hand, the Erlang assumption can lead to explicit expressions for ruin related functions due to its mathematical tractability. On the other hand, the constant inter-review time can be approximated by the Erlangization techniques proposed by Asmussen et al. [3]. (see also e.g. [11,12,1,2,16]). Suppose that the solvency is discretely monitored, and define the ruin time by τ = Z J , where J = inf{j ≥ 1 : U Zj < 0}. Let τ = ∞ if U Zj ≥ 0 for all j = 1, 2, · · · . Let δ ≥ 0 be the interest force, and let I(A) be the indicator function of the event A. We are interested in the Gerber-Shiu function where w defined on [0, ∞) × (0, ∞) is a penalty function of the surplus prior to ruin and the deficit at ruin.
The remainder of this paper is organized as follows. In Section 2, we derive the discounted density of the surplus increment. The Laplace transforms of the Gerber-Shiu function is given in Section 3. Finally, when the claim size density is a combination of exponentials, we find the explicit expressions for the Laplace transforms of the ruin time.

2.
Preliminaries. For q > 0, let e q denote an exponential r.v. with mean 1/q (independent of other random quantities), and let e q,1 , e q,2 , · · · be independent copies of e q . For and let f q,n,i (x) be the corresponding density function. Since Y i (t) has stationary independent increments property, we have for n ≥ 2 which gives a recursive approach to compute f q,n,i with the starting point f q,1,i . For convenience, we put in the remainder of this paper. Note that f + q,n,i and f − q,n,i are defined on (0, ∞). First, let us consider the case n = 1. For i = 1, 2, let be the Laplace exponent of Y i , and for any q > 0, let ρ q,i be the positive root of equation ψ i (s) = q. Note that ρ q,i is unique and it is the right inverse of ψ i . It follows from Corollary 8.9 in [8] that By (5) we immediately obtain where a q,n,i,j 's are some constants defined in the following proof.
3. Analysis of the Gerber-Shiu functions. First, we derive integral equations for the Gerber-Shiu functions. Suppose that η 1 = c 1 . By conditioning on the time of the first review time Z 1 and the surplus increment U Z1 − U 0 , we have where (14) where ω 2 (u) = ∞ u w(u, x − u)f − γ+δ,n,2 (x)dx. Next, we derive Laplace transforms of the Gerber-Shiu functions from the integral equations (12) and (14). In the sequel, we denote the Laplace transform of a function by adding a hat on the corresponding letter. For example, a n,i,j = a γ+δ,n,i,j , i = 1, 2. By Lemma 1, we can write (12) as follows,
By (18) and (19) one easily obtainŝ andφ The Laplace transformsφ 1 (s) andφ 2 (s) can be determined by (20) and (21), provided that we can determine the unknown constants, A 1,k , A 2,k , k = 1, · · · , n. In order to determine A 1,k 's and A 2,k 's, we consider the roots of the following equation,   n j=1 a n,1,j n j=1 a n,2,j (ρ 2 − s) j = 0. Proof. First, it is easily seen that ρ 1 and ρ 2 are not roots of equation (22). Hence, (22) is equivalent to the following equation Now we prove that equation (23) has the same number of roots on the right half complex plane as the following equation To this end, we use Rouche's theorem. It is easily seen that both sides of (23) are analytic function of s. Let r > 0 be a sufficiently large number, and denote by C r the contour containing the imaginary axis running from −ir to ir and a semicircle with radius r running clockwise from ir to −ir.

ZHIMIN ZHANG, YANG YANG AND CHAOLIN LIU
For s on the semicircle, we have: as r → ∞, f − n,i (s) → 0, n j=1 Hence, when the radius r is sufficiently large, we know that the module of the left side of (23) is larger than that of the right side.
Remark 1. The proof of Theorem 1 relies on Rouché's theorem, which requires that there are no roots on the boundary of some domains. However, for δ = 0 the suitable domain can not be constructed. As a result, we use the generalized Rouche's theorem proposed in [7]. See e.g. [15]. Proof. When δ = 0, (22) reduces to   n j=1 a n,1,j , then (27) has a root s = 0. For convenience, we put On the open disk D = {z ∈ C : |z| < 1} and its boundary ∂D = {z ∈ C : |z| = 1}, we define where R is a sufficiently large positive number. Immediately we have is equal to that of θ 1 (s) = 0 minus 1. Using the same arguments used in Theorem 1, we can show that equations θ 1 (s) = 0 and (ρ 1 − s) n (ρ 2 − s) n = 0 have the same number of roots on the right half complex plane. Since the latter has 2n positive roots (counting multiplicity), the former also has 2n roots with positive real parts. Hence, θ 1 (s) − θ 2 (s) = 0 has 2n − 1 roots with positive real parts. This completes the proof.
Upon Laplace inversion we get which are general formulas for the Laplace transforms of the ruin time under the claim size density assumption (32). We present some numerical examples. Let us consider the following three claim size distributions (1) a sum of two exponentials with mean 1/3 and 2/3; (2) an exponential distribution with mean 1; (3) a mixture of two exponentials: one exponential with mean 2 (mixing probability 1/3) and one exponential with mean 1/2 ( mixing probability 2/3). The corresponding claim size densities take the form of combination of exponentials: . Moreover, we find that the common mean of these three distributions is 1, while the variances are 0.56, 1 and 2, respectively. See [2].
Assume that the inter-review times are Erlang(2) distributed with density function f T (t) = te −t , t ≥ 0. Set c 1 = 1.1, c 2 = 2, σ 2 = 0.5. Since the mean of claim size is 1, the safety loading condition c 1 > λµ X holds true. Furthermore, set δ = 0 and w ≡ 1, then the Gerber-Shiu functions reduce to ruin probability. We can use (44) to compute φ i (u), i = 1, 2. We plot the ruin probability curves in Figure 1. It follows that both φ 1 (u) and φ 2 (u) are decreasing functions of the initial surplus level. Furthermore, we observe that φ 1 (u) > φ 2 (u), which is possibly due to that ruin is more likely to occur when the initial effective premium rate η 1 = c 1 because of c 1 < c 2 . In Tables 1-3, we illustrate some exact values of ruin probabilities.
Comparing the values of the same cells across these tables, we find that the ruin probabilities appear to increase with the variance of the claim size distribution, which implies that the claim with larger variance has higher risk for the insurer.