 Advances in Pure Mathematics, 2011, 1, 54-58 doi:10.4236/apm.2011.13012 Published Online May 2011 (http://www.scirp.org/journal/apm) Copyright © 2011 SciRes. APM On a Class of Dual Model with Divided Threshold Yuzhen Wen School of Mathemat i cal Sci ences, Qufu Normal University, Qufu, China E-mail: wenyzhen@163.com Received January 24, 2011; revised March 28, 2011; acce pted April 1, 2011 Abstract In this paper, we consider the dual of the generalized Erlang (n) risk model under a threshold dividend strat-egy. We derive an integro-differential equation satisfied by the expectation of the discounted dividends until ruin. The case when profits follow an exponential distribution is solved. Keywords: Threshold Strategy, Dual Risk Model, Generalized Erlang (n) Risk Process 1. Introduction In recent years, a few interesting results have been obtained on a model which is dual to the classical insurance risk model. See Avanzi et al. Avanzi and Gerber [2,3] and A. C. Y. Ng  for example. In this model, the surplus at time t is  ()1,0NtkkUtuctZuctSt t  . (1) where u and c are constants, u is the initial sur-plus, 0c is the rate of expenses, 1NtkkSt Z is the aggregate positive gains process and :1,2,kZk is a sequence of independent and identically distributed claim amount nonnegative random variables with a com- mon probability density function ,0py y. The or-dinary renewal process ,0Nt t denotes the num-ber of gains up to time t with 12max 1:kNtkW WWt where the i.i.d gains waiting times iW have a common generalized Erlang (n) distribution, i.e. the iWs are distributed as the sum of n independent and exponen-tially distributed random variables: 12 ,1,2,,,inWin  (2) where 1, 2,,jjn may have different exponential parameters 0.j Furthermore, we assume that 1iiW and 1iiZ are independent. In this model, the expected increase of the surplus per unit time is  11EX cEW and is assumed to be positive. In this model, the premium rate is negative, causing the surplus to decrease. Claims, on the other hand, cause the surplus to increase. Thus the premium rate should be viewed as an expense rate and claims should be viewed as profits or gains. Though not very popular in insurance mathematics, this model has appeared in various litera- ture (see Cramer , Seal , Takcs  and the refer-ences cited therein. In Avanzi et al. , the authors stud-ied the expected total discounted dividends until ruin for the dual model under the barrier strategy by means of integro-differential equations. In  the authors consider a Sparre Andersen risk process that is perturbed by an independent diffusion process in which claim inter-arrival times have a generalized Erlang (n) distribution. In this paper, we will study the expectation of the dis-counted dividends until ruin. We get integro-differential equation of the expectation of the discounted dividends until ruin. We also get the the expectation of the dis-counted dividends until ruin when profits follow an ex-ponential distribution. 2. Main Result We now consider a threshold dividend strategy. When Ut is below b, no dividends are paid and the surplus decreases at the original rate 1c. When Ut is above b, the surplus would decrease at a different rate 21cc and dividends are paid at rate 21cc Then Ut can be expressed by  21dd ,;ddd ,0;ct StUt bUt ctStb Ut  0t Let inf0 :0TtUt Y. Z. WEN Copyright © 2011 SciRes. APM 55(T if ruin does not occur) be the time of ruin and IA be equal to 1 if event A occurs and 0 otherwise. The total discounted dividends until ruin is  210() dTtDbcceIUtbt  where 0 is the force of interest for valuation. Let  ;0Vub EDbUu denote the expectation of the discounted dividends until ruin, if the threshold dividend strategy with parameter b is applied. Since Ut have different paths for 0Ut b and Ut b, we define  12;, 0,;;, .Vubb uVub Vubub The following theorem provides integro-differential equations for the expectation of the discounted dividends of ;Vub. Theorem 2.1 The expectation of the discounted divi- dends of ;Vub satisfy the following integro-diffe- rential equations: when 0ub we have  111120d1;d;d ;d;niiibubucIVubuVu ybpy yVu ybpy y  (3) when ub, we have  221122101d1;d;dniiinnjiji jcIVubuVu ybpyycc  (4) where I is the identity operator. Proof. Let 00S and 12jjS  for 1, 2,,1jn Define  ,;,0ij jVub EDbS tUu with  ,0iiVuVu for 1, 2i. We first consider the case when ub. We consider the infinitesimal interval from jS to d.jSt For 0,1, ,2jn, we have 2,12, 212,1221;dd;dd;1.dtjjjjjdtVube PtEVuctbPtEVuctbecc   (5) Note that d1d dtetot . Also we have   11112, 22,2, 2d1dd,ddd,d;d;;dddjjjjjjjPt totPttotEVuc tbVubVubc totu    Substituting these formulas into (5), subtracting 2, ;jVub from both sides, interpreting dt and dot terms, canceling common factors and letting d0t, we have  12, 1122,21;d;djjjjVubIcVubccu  (6) for 0,1,,2jn. Similarly for 1jn, we have 22,12210d;d;d .nnnIcV ubuVu ybpyycc  (7) Thus we have  221122101d1;d;dniiinnjiji jcIVubuVu ybpyycc  Now suppose 0ub. Similar arguments as above shows that we have 1,d11,111,11;dd;dd;jtjjjjVubeP tEVuctbPtEVuctb  (8) for 0,1,,2jn and  11,1120d;d;d ;dnnbunnbuIcVubuVu ybpy yVu ybpy y (9) for 1jn respectively. Substituting (8) into (9), we have (3). □ Remark 2.2 Consider a compound Poisson dual mod-el, i.e. the iW has an exponential distribution with pa-rameter . When 0ub, we have  111120d;;d;d ;d.bubuVubVub cuVu ybpy yVu ybpy y  Y. Z. WEN Copyright © 2011 SciRes. APM 56 When ub, we have  2222210d;;d;dVubVub cuVuybpyyc c  Thus Theorem 2.1 generalized results obtained in A. C. Y. Ng . Corolla ry 2.3 When iWs have generalized Erlang (2) distributions, we have  11122 11120dd11 ;dd;d ;dbubuccIIVubuuVu ybpy yVu ybpy y      (10) for 0ub and  22222 11221022121dd11 ;dd1;d11ccIIVubuuVuybpyyc ccc        (11) for ub with the boundary conditions: 10; 0Vb (12) 120; 0;Vbb Vbb  (13) 21201021d; d;||ddub ubVub Vubccccuu  (14)  111022 112222201121 21221dd11 ;|ddd1dd1;|d11(1 ).ububccIIVubuucIucIVubucc cc        (15) Proof. Since ruin is immediate when 0u, we have (12) and (13) by the continuity condition, According to L. J. Sun  and Y. H. Dong et al. , we have  21 11;;Vub Vub. This together with (6) and (8) yields (14). Similarly we can get (15) from (3) and (4). □ Example 2.4 (Expectation of Discounted Dividends when Profits Follow an Exponential Distribution) Let profits follow an exponential distribution with py ye for 0y. Putting the distribution function into (11) for ub, we have  22222 11221022121dd11 ;dd1;d11yccIIVubuuVuybeycccc        Applying the operator ddIu to both sides, we get  22222dd ;dd;IcIVubuuVub   It follows that we have 3222212342532ddd 0dddVVVBBBBVBuuu where 22112222 221221 122231221124125212112,11 ,111 1,11 ,11 .cBcc cBccBBBcc           The third-order linear differential equation above has a particular solution 21cc. Since the characteristic equa-tion of the differential equation 3212340BrBrBr B has two negative roots 1r and 2r and a positive root 3r, we have 312 212123;ruru ruccVub DeDeDe  where 1D, 2D and 3D are constants. Similar to An-drew C.Y. Ng , we have 10D, 20D and 30D. Hence we have 1221212;ruru ccVub DeDe  Y. Z. WEN Copyright © 2011 SciRes. APM 57We put the distribution function of ypy e into (10). Then, for 0bu, we have 11122 11102dd11 ;dd;d;d.bu yybuccIIVubuuVu ybeyVu ybey      (16) Applying the operator ddIu to both sides, we get 321111234132ddd 0dddVVVBBBBVuuu  where 21112211 121221 12113122112412,11 ,1111,11.cBcc cBccBB   Hence we have 3121123;susu suVub EeEeEe where 1E, 2E and 3E are constants, 1s, 2s and 3s 12 30sss are the solutions of the character-istic equation 3212340BsBsBs B  Since 0, 0Vb, we get 1230.EEE (17) Substituting back the solution for 1;Vub and 2;Vub into (16), we have  312121122 1112 31112212dd11dd;d.susu suburbuyuurb buccIIuuEe EeEeDeVybey erDcceer        Since the expression above must be satisfied for all 0ub, the sum of the coefficients of ue must be zero. Thus we have 312123121231221120sbsb sbrb rbEeEe EesssDeDec crr  (18) On the other hand, since 120; 0;Vbb Vbb , we have 3121212 32112susu surb rbEe EeEeccDe De  (19) It follows from (10) and (11) that we have  312121122 1112 32222 11211221 21221dd11dddd11dd11(1 ).susu suru ruccIIuuEe EeEeccIIuuccDe Decc cc                    (20) Since  2211210;0; ,cVbbcV bbcc we have 12312211 22111223321.rbrbsbsb sbcrDe rDecsEesEesEecc (21) From Equations of (17), (18), (19), (20) and (21), we can get the solution of 1;Vub and 2;Vub. 3. Acknowledgements This work was supported by National Natural Science Foundation of China (No. 10771119). The author would like to thank Professor Chuancun Yin for his support and useful discussions. 4. References  B. Avanzi, H. U. Gerber and E. S. W. 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