行政院國家科學委員會專題研究計畫 成果報告
在 Erlang(2) 風險過程下貼現分配函數之研究
計畫類別： 個別型計畫 計畫編號： NSC93-2416-H-002-048- 執行期間： 93 年 08 月 01 日至 94 年 07 月 31 日 執行單位： 國立臺灣大學財務金融學系暨研究所 計畫主持人： 蔡啟良 報告類型： 精簡報告 處理方式： 本計畫可公開查詢中 華 民 國 94 年 10 月 12 日
行政院國家科學委員會補助專題研究計畫
□ 成 果 報 告
□期中進度報告
在 Erlang(2) 風險過程下貼現分配函數之研究
計畫類別：□ 個別型計畫
□ 整合型計畫
計畫編號：NSC 93－2416－H－002－048－
執行期間：93 年 08 月 01 日至 94 年 07 月 31 日
計畫主持人：
蔡 啟 良
共同主持人：
計畫參與人員：
成果報告類型(依經費核定清單規定繳交)：□精簡報告
□完整報告
本成果報告包括以下應繳交之附件：
□赴國外出差或研習心得報告一份
□赴大陸地區出差或研習心得報告一份
□出席國際學術會議心得報告及發表之論文各一份
□國際合作研究計畫國外研究報告書一份
處理方式：除產學合作研究計畫、提升產業技術及人才培育研究計畫、
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□涉及專利或其他智慧財產權，□一年□二年後可公開查詢
執行單位：
國立台灣大學 財務金融學研究所
中
華
民
國
94
年
10
月
11
日
中文摘要
在此計畫中，我將以傳統的連續時間風險模型之保險公司的盈餘過程為基礎，在等待時間 (即兩個相鄰保險理賠的間隔時間)服從獨立且相同之爾朗(2)分配的假設下，推導破產前一 瞬間的盈餘和破產時的不足額之貼現聯合和邊際分配和密度函數，以及導致破產的保險理 賠之貼現聯合和邊際分配和密度函數，並證明這些邊際分配和密度函數分別滿足一瑕更新 方程式。此外，我想嘗試探討這些分配和密度函數分別在不同爾朗(2)和爾朗(1)的假設下 是否有類似性。最後，我將尋找一些穩定而有效率的數值方法，在期初盈餘給定和保險理 賠服從指數分配的假設下，計算這些邊際分配和密度函數的值，做為保險公司和監理機關 的參考。 關鍵詞 : 盈餘過程，爾朗(2)過程，破產時間，貼現分配函數，瑕更新方程式Abstract
In this project, we will try to derive explicit expressions for the discounted joint and marginal distribution and probability density functions of the surplus immediately prior to the time of ruin and the deficit at the time of ruin, and for the discounted distribution and probability density function of the amount of the claim causing ruin, based on Erlang(2) assumption. Also, we will want to show that these distribution and probability density functions satisfy defective renewal equations. Moreover, we hope we can find the similarity between expressions for these functions based on Erlang(1) and Erlang(2) processes, respectively. Finally, we would like to seek for some stable and effective numerical methods of computing the expressions for the discounted
distribution and density functions, and get the numerical values for the insurer’s and regulator’s information if we the claim size is exponentially distributed.
Keywords : Surplus process; Erlang(2) process; Time of ruin; Discounted distribution function; Defective renewal equation.
計畫緣由與目的
Consider the surplus process of the insurer in the classical continuous time risk model,
(1) U(t)uct S(t), t≧0
where U(t) is the surplus of the insurer at time t, u=U(0) is the initial surplus, c is the constant rate per unit time at which the premiums are received, S(t) = X1+ X2+…+ XN(t)(with S(t) = 0 if N(t)=0) denotes the aggregate claims up to time t, N(t) is the number of claims up to time t, and the individual claim sizes X1, X2,…,independent of N(t), are positive, independent and
identically
distributed random variables with common distribution function P(x)=Pr(X ≦x) and mean p1. Let {Ti: i = 1, 2, …} be a sequence of independent and identically distributed random variables representing the times between claims (T1is the time until the first claim) with common
probability density function k(t); then an Erlang(2) process is the process where k(t)=λ2t exp(-λt), t>0 (that is, a Gamma(2,λ) distribution). In the traditional surplus process, k(t) is most assumed exponentially distributed with mean 1/λ (that is, Erlang(1) or Gamma(1, λ)), which is equivalent to that N(t) is a Poisson distribution with parameterλ; in this case, c=λ p1(1+θ) whereθ>0 is the relative security loading. For Erlang(2) assumption, the constant premium rate c becomes c=[E(X1)/E(T1)](1+θ)=(λp1/2)(1+θ) (see Willmot and Dickson (2003) for more details). There are some articles discussing ruin problems based on Erlang(2) process, for example, Dickson (1998), Dickson and Hipp (1998) and (2001), and Cheng and Tang (2003).
Let T = inf{t:U(t)<0} be the time of ruin. When ruin occurs at time T, |U(T)| (or -U(T)) is called the severity of ruin or the deficit at the time of ruin, U(T-) (the left limit of U(t) at t=T) is called the
surplus immediately before the time of ruin, and [U(T-)+|U(T)|] is called the amount of the claim causing ruin. Let f(x,y,t|u) denote the joint probability density function of U(T-), |U(T)| and T. Define the discounted joint and marginal probability density functions of U(T-) and |U(T)| based on model (1) with the discount factor δ≧ 0 as follows:
, ) | , , ( ) | ; , ( 0 e f x y t u dt u y x f
t , ) | , , ( ) | ; , ( ) | ; ( 0 0 0 1
f x y u dy e f x y t u dtdy u x f t and . ) | , , ( ) | ; , ( ) | ; ( 0 0 0 2
f x y u dx e f x y t u dtdx u y f tWhen δ=0, the probability density functions are denoted by f(x,y;0|u), f1(x;0|u), and f2(y;0|u), respectively. There have been many papers studying the marginal and joint distributions of T, U(T-) and |U(T)|, based on Erlang(1) assumption. See Gerber, Goovaerts and Kaas (1987),
and Shiu (1997) and (1998), Willmot and Lin (1998), Lin and Willmot (1999), Wang and Wu (2000), Tsai (2001), and references therein. Dickson (1992) proposed an explicit expression for f1(x;0|u); later, Gerber and
Shiu (1998) extended Dickson's formula for the case δ≧0 and got f1(x;δ|u) by a martingale approach; Lin and Willmot (1999), however, used some different methods to derive explicit expressions for f(x,y;δ|u), f1(x;δ|u), and f2(y;δ|u); Tsai (2001) further gave explicit expressions for the discounted probability density and distribution functions based on (1) with an independent diffusion factor added.
Based on Erlang(2) assumption, Cheng and Tang (2003) derived an expression for f(x,y;0|u). However, the derived joint probability density function and its implied marginal probability density functions can not be compared with these in Lin and Willmot (1999) and Tsai (2001) both based on Erlang(1) process, and hence the relation between expressions for these functions based on Erlang(1) and Erlang(2) processes, respectively, can not be observed.
In this project, we derive explicit expressions for the discounted joint and marginal distribution functions of the surplus immediately prior to the time of ruin and the deficit at the time of ruin, and for the discounted distribution function of the amount of the claim causing ruin, for the Erlang(2) process. We also show that these distribution functions satisfy defective renewal equations. Moreover, we find an important similarity between expressions for these functions based on the Erlang(1) and Erlang(2) processes, respectively. When some condition holds, the expressions for Erlange(2) would reduce to those which have the same function forms as corresponding ones for the Erlang(1) process.
計畫成果自評
This project has been rewritten as a paper, submitted to and appeared in a well-known actuarial journal, Insurance: Mathematics and Economics (SSCI).
Tsai, Cary Chi-Liang, and Li-Juan Sun. 2004.
"On the Discounted Distribution Functions for the Erlang(2) Risk Process", Insurance: Mathematics and Economics 35: 5-19.
參考文獻
Cheng, Y., Tang, Q., 2003. Moments of the surplus before ruin and the deficit at ruin in the Erlang(2) risk process. North American Actuarial Journal, 7, (1), 1-12.
Dickson, D.C.M., 1992. On the distribution of the surplus prior to ruin. Insurance: Mathematics and Economics 11, 191-207.
Dickson, D.C.M., 1993. On the distribution of the claim causing ruin. Insurance: Mathematics and Economics 12, 143-154.
Dickson, D.C.M., 1998. On a class of renewal risk processes. North American Actuarial Journal, 2 (3), 60-73.
Dickson, D., Egidio dos Reis, A. and Waters, H., (1995). Some stable algorithms in ruin theory and their applications, ASTIN Bulletin, 25, 153-175.
Dickson, D.C.M., Hipp, C., 1998. Ruin probabilities for Erlang(2) risk process. Insurance: Mathematics and Economics, 22, 251-262.
Dickson, D.C.M., Hipp, C., 2001. On the time to ruin for Erlang(2) risk processes. Insurance: Mathematics and Economics, 29, 333-344.
Dickson, D.C.M., Waters, H., 1992. The probability and severity of ruin in finite and in infinite time. ASTIN Bulletin 22, 177-190.
Dufresne, F., Gerber, H.U., 1988. The surpluses immediately before and at ruin, and the amount of the claim causing ruin. Insurance: Mathematics and Economics 7, 193-199.
Gerber, H.U., Goovaerts, M., Kaas, R., 1987. On the probability and severity of ruin. ASTIN Bulletin 17, 151-163.
Gerber, H.U., Shiu, E.S.W., 1997. The joint distribution of the time of ruin, the surplus
immediately before ruin, and the deficit at ruin. Insurance: Mathematics and Economics 21, 129-137.
Gerber, H.U., Shiu, E.S.W., 1998. On the time value of ruin. North American Actuarial Journal 2 (1), 48-78.
Lin, X., Willmot, G.E., 1999. Analysis of a defective renewal equation arising in ruin theory. Insurance: Mathematics and Economics 25, 63-84.
Sun, L.J., 2002. On the discounted penalty at ruin in the Erlang(2) risk process. Technical report, Tsinghua University, China.
Tsai, C.C.L., 2001. On the discounted distribution functions of the surplus process perturbed by diffusion. Insurance: Mathematics and Economics 28, 401-419.
Tsai, C.C.L., Willmot, G.E., 2002. A generalized defective renewal equation for the surplus process perturbed by diffusion. Insurance: Mathematics and Economics 30, 51-66. Wang, G., Wu, R., 2000. Some distributions for classical risk process that is perturbed by
diffusion.
Willmot, G.E., Dickson, D.C.M., 2003. The Gerber-Shiu discounted penalty function in the stationary renewal risk model. Insurance: Mathematics and Economics 32, 403-411.
Willmot, G.E., Lin, X., 1998. Exact and approximate properties of the distribution of the surplus before and after ruin. Insurance: Mathematics and Economics 23, 91-110.
