Studies on a Double Poisson-Geometric Insurance Risk Model with Interference

Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2013, Article ID 128796, 8 pages http://dx.doi.org/10.1155/2013/128796 Research Article Studies on a Double Poisson-Geometric Insurance Risk Model with Interference Yujuan Huang1 and Wenguang Yu2 1 School of Science, Shandong Jiaotong University, Jinan 250023, China 2 School of Insurance, Shandong University of Finance and Economics, Jinan 250014, China Correspondence should be addressed to Yujuan Huang; [email protected] Received 11 January 2013; Accepted 5 March 2013 Academic Editor: Hua Su Copyright © 2013 Y. Huang and W. Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper mainly studies a generalized double Poisson-Geometric insurance risk model. By martingale and stopping time approach, we obtain adjustment coefficient equation, the Lundberg inequality, and the formula for the ruin probability. Also the Laplacetransformationofthetimewhenthesurplusreachesagivenlevelforthefirsttimeisdiscussed,andtheexpectationandits variance are obtained. Finally, we give the numerical examples. 1. Introduction Upadhye [6] study the convolution of compound negative binomial distributions with arbitrary parameters. The exact In insurance mathematics, the classical risk model has been expression and also a random parameter representation are () the center of focus for decades [1–3]. The surplus in the obtained. Cossette et al. [7]presentacompoundMarkov classical model at time canbeexpressedas binomial model, which is an extension of the compound 1() binomial model. The compound Markov binomial model is based on the Markov Bernoulli process which introduces () =+− ∑ , (1) =1 dependency between claim occurrences. Recursive formulas are provided for the computation of the ruin probabilities where =(0)>0is the initial capital, >0is the constant over finite- and infinite-time horizons. A Lundberg exponen- { (), ≥ 0} rate of premium, and 1 is a Poisson process, with tial bound is derived for the ruin probability, and numerical >0 Poisson rate 1 denoting the number of claims up to examples are also provided. Yang and Zhang [8]investigate ,,... time . The individual claim sizes 1 2 , independent of a Sparre Andersen risk model in which the inter-claim times { (), ≥ 0} 1 , are independent and identically distributed are generalized Erlang(n) distributed. Czarna and Palmowski nonnegative random variables with common distribution 2 [9] focus on a general spectrally negative Levy insurance function () with mean ,variance, and moment risk process. For this class of processes, they analyze the so- generating function () = [ ]. called Parisian ruin probability, which arises when the surplus But in the Poisson process, the expectation and variance process stays below 0 longer than a fixed amount of time >0. are equal. This is obviously not consistent with actual situa- In this paper, we will consider a double Poisson- tion. So recently there is a huge amount of literature devoted Geometric risk model with diffusion in which the arrival to the generalization of the classical model in different ways. of policies is a Poisson-Geometric process and the claims Lu and Li [4] consider a Markov-modulated risk model in process follows the compound Poisson-Geometric process. which the claim interarrivals, claim sizes, and premiums are For more details and new developments on the Poisson- influenced by an external Markovian environment process. Geometric risk model, the interested readers can refer to [10– Tan and Yang [5] discuss the compound binomial risk model 13]. with an interest on the surplus under a constant dividend The rest of the paper is organized as follows. In Section 2, barrier and periodically paying dividends. Vellaisamy and the risk model is introduced. In Section 3,weobtainthe 2 Discrete Dynamics in Nature and Society adjustment coefficient equation and the formula of ruin prob- For the risk model (3), the time to ruin, denoted by ,is ability.Thenwepresenttheeffectoftherelatedparameterson defined as the adjustment coefficient. In Section 4, using the martingale method, the time when the surplus reaches a level firstly is =inf {≥0|() <0} . (7) considered, and the expectation and its variance are obtained. Numerical illustrations are also given. And define the ruin probability with an initial surplus >0 by (),namely, 2. The Risk Model () = Pr (<∞|(0) =) . (8) Definition 1 (see [10]). A distribution is said to be Poisson- (, ) Geometric distributed, denoted by , if its generating 3. The Ruin Probability function is (−1) {(); ≥0} { }, Define the profits process by ;thatis, exp 1− (2) 3() where >0, 0≤<1.Notethatif=0,then () =2 () − ∑ +() . (9) the Poisson-Geometric distribution degenerates into Poisson =1 distribution. Obviously we have Definition 2 (see [10]). Let >0and 0≤<1,then {(), ≥0} is said to be a Poisson-Geometric process with 2 3 parameters , if it satisfies [ ()] =[ − ], 1−2 1−3 (1) (0) =; 0 [ ()] =2 [ ()]+ [ ()]⋅2 [ ] (2) {(), ≥0} has stationary and independent incre- Var Var 2 Var 3 ments; 2 +[3 ()]⋅Var []+ Var [ ()] (3) for all >0, () is a Poisson-Geometric distributed (10) [()] = /(1 − 2 2 with parameters , ,and 2 (1 + 2) 3 (1 + 3) 2 =[ + ), Var[()] = (1 + )/(1 −) . 2 2 (1 − 2) (1 − 3) The corresponding moment generating function of () () = [( − 1)/(1 − )] 2 is () exp . + 3 +2]. Then the double Poisson-Geometric risk model with 1−3 interference is defined as 3() Let () =+ () − ∑ +() , (3) 2 =1 = 2 − 3 , 1− 1− where 2() is the number of premium up to time and 2 3 follows a Poisson-Geometric distribution with parameters 2 2 2 2 (11) 2 (1 + 2) 3 (1 + 3) 3 2 and 2; 3() is the number of claims up to time and follows = + + + . 2 2 1− a Poisson-Geometric distribution with parameters 3 and 3. (1 − 2) (1 − 3) 3 () is the standard Brownian motion and is a constant, representing the diffusion volatility parameters. Throughout Then this paper, we assume that 2(), 3(), (),and{} are =, mutually independent. [ ( )] In order to ensure the insurance company’s stable opera- (12) Var [ ()] =. tion, we assume () 3 Lemma 3. The profits process {(); ≥0} has the following [ () − ∑ +()]>0, 2 (4) properties: =1 which implies (1) (0) =; 0 2 3 (2) {(); ≥0} has stationary and independent incre- − >0. (5) 1−2 1−3 ments. Let Theorem 4. For the profits process {(); ,thereisa ≥0} 2 3 function () such that = (1+) . (6) 1−2 1−3 [−()]=() . Then >0is the relative security loading factor. (13) Discrete Dynamics in Nature and Society 3 0.164 0.855 0.85 0.162 0.845 0.16 0.84 0.158 0.835 0.83 0.156 0.825 0.154 0.82 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.44 0.45 2 Figure 1: The impact of on . Figure 2: The impact of 2 on . Proof. Consider 0.89 −() 0.88 [ ]={exp [−2 ()]} ⋅ 0.87 3() 0.86 ×{exp [ ∑ ]} ⋅ exp { [− ()]} =1 0.85 (− −1) (14) 0.84 = { [ 2 exp − 1−2 0.83 0.82 () −1 1 2 2 +3 + ]}. 0.81 1−3 () 2 0.8 Let (− −1) () −1 1 0.79 () = 2 + + 22. 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 − 3 (15) 1−2 1−3 () 2 3 Then we obtain (13). Figure 3: The impact of 3 on . Theorem 5. Equation () =0 (16) which imply has a unique positive solution =>0,and(16) is said to be an adjustment coefficient equation of the risk model (3) and 2 3 3 >0 (0) =− + =− <0. (18) is said to be an adjustment coefficient. 1−2 1−3 1−3 Proof. From (15), we have (0) =, 0 and since It is easy to see that the moment generating function () 0< <1 − (1 − ) [] is an increasing function. Due to 3 ,there 2 (2 −1) 3 3 2 () = + + , exists an 1 such that (1)=1/3;thatis,1−3() > 0 − 2 2 (1 − 2 ) (1 − 3 ()) when 0<<1.Sowhen0<<1, () > 0 and () is a convex function with lim→+∞() =.Then +∞ 2 − (1 − )(1+ −) () () = 2 2 2 itcanbeshownthat has a unique positive solution on − 3 (0, +∞) (1 − 2 ) . = 0.5 = 0.4 = 0.2 =0.9 3 (1 − 3)[1−3 ()] Example 6. Suppose , 2 , 3 , 2 ,and + = 0.6 = 0.9 = 1.4 [1 − ()]4 3 , , .By(16), we obtain the adjustment 3 coefficient = 0.158. Moreover, we give the effect of related 2 parameters on adjustment coefficient ; see Figures 1, 2, 3, 4, ×{(1− ())[2]+ 2[ ()] }+2, 3 5, 6,and7. (17) For the profits process {(); ≥0},let ={(V); V ≤}. 4 Discrete Dynamics in Nature and Society 0.86 0.846 0.855 0.844 0.842 0.85 0.84 0.845 0.838 0.84 0.836 0.834 0.835 0.832 0.83 0.83 0.88 0.89 0.9 0.91 0.92 0.93 0.94 0.95 0.86 0.87 0.88 0.89 0.9 0.91 0.92 0.93 0.94 0.95 2 Figure 4: The impact of 2 on . Figure 6: The impact of on . 0.86 0.87 0.855 0.86 0.85 0.845 0.85 0.84 0.84 0.835 0.83 0.83 0.825 0.82 0.82 0.57 0.58 0.59 0.6 0.61 0.62 0.63 0.81 1.36 1.37 1.38 1.39 1.4 1.41 1.42 1.43 1.44 1.45 3 Figure 5: The impact of 3 on . Figure 7: The impact of on . Theorem 7. {(); ; ≥ 0}is a martingale, where () = −(+()) () Proof. Consider / . −()− −()−() [ |V]=[ |V] Proof. Consider =[−(V)−()−[()−(V)]−(−V)() |] −(+()) V [ () |]=[ |] V () V −(V)−() −[()−(V)]−(−V)() = ⋅[ |V] −(+(V)) −(()−(V)) −(V)− =[ |] = .

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