The Expected Discounted Tax Payments on Dual Risk Model under a Dividend Threshold *

In this paper, we consider the dual risk model in which periodic taxation are paid according to a loss-carry-forward system and dividends are paid under a threshold strategy. We give an analytical approach to derive the expression of gδ(u) (i.e. the Laplace transform of the first upper exit time). We discuss the expected discounted tax payments for this model and obtain its corresponding integro-differential equations. Finally, for Erlang (2) inter-innovation distribution, closedform expressions for the expected discounted tax payments are given.


Introduction
Consider the surplus process of an insurance portfolio which is dual to the classical Cramér-Lundberg model in risk theory that describes the surplus at time , where is the initial capital, the constant is the rate of expenses, and   is aggregate profits process with the innovation number process being a renewal process whose inter-innovation times i Λ have common distribution F .We also assume that the innovation sizes , independent of i , forms a sequence of i.i.d.exponentially distributed random variables with exponential parameter .There are many possible interpretations for this model.For example, we can treat the surplus as the amount of capital of a business engaged in research and development.The company pays expenses for research, and occasional profit of random amounts arises according to a Poisson process.
Due to its practical importance, the issue of dividend strategies has received remarkable attention in the literature.De Finetti [1] considered the surplus of the com-pany that is a discrete process and showed that the optimal strategy to maximize the expectation of the discounted dividends must be a barrier strategy.Since then, researches on dividend strategies has been carried out extensively.For some related results, the reader may consult the following publications therein: Bühlmann [2], Gerber [3], Gerber and Shiu [4,5], Lin et al. [6], Lin and pavlova [7], Dickson and Waters [8], Albrecher et al. [9], Dong et al. [10] and Ng [11].Recently, quite a few interesting papers have been discussing risk models with tax payments of loss carry forward type.Albrecher et al. [12] investigated how the loss-carry forward tax payments affect the behavior of the dual process (1.1) with general inter-innovation times and exponential innovation sizes.More results can be seen in Albrecher and Hipp [13], Albrecher et al. [14], Ming et al. [15], Wang and Hu [16] and Liu et al. [17,18].Now, we consider the model (1.1) under the additional assumption that tax payments are deducted according to a loss-carry forward system and dividends are paid under a threshold strategy.We rewrite the objective process as


. that is, the insurance company pays tax at rate   0,1   on the excess of each new record high of the surplus over the previous one; at the same time, dividends are paid at a constant rate  whenever the surplus of an insurance portfolio is more than b and otherwise no dividends are paid.Then the surplus process of our model . where 1 is the indicator function of event A and is the surplus immediately before time .t For practical consideration, we assume that the positive safety loading condition It needs to be mentioned that we shall drop the subscript  whenever  is zero.
The rest of this paper is organized as follows.In Sec-tion 2, We derive the expression of u the Laplace transform of the first upper exit time).We also discuss the expected discounted tax payments for this model and obtain its satisfied integro-differential equations.Finally, for Erlang (2) inter-innovation distribution, closed-form expressions for the the expected discounted tax payments are given.

Main Results and Proofs
,0, u u  denote the Laplace trans- form of the upper exit time 0 , which is the time for the first time without leading to ruin before that event.In is the probability that particular, For general innovation waiting times distribution, one can derive the integral equations for g u u u b 2) , and By changing variables in from Equation (2.6) and from Equation (2.7), we have for 0

for
, and Then, differentiating both sides of from Equation (2.8) and from Equation (2.9) with respect to , one gets , and 12) for , and 1, 2, , 1 Again, by changing variables in Equation (2.12) and Equation (2.13) and then differentiating them with respect to , we obtain for In addition, the boundary conditions for   , V u b are as follows: With the preparations made above, we can now derive analytic expressions of the expected -th moment of the accumulated discounted tax payments for the surplus We claim that the process process : lim g before that event.In particular, u g u

  
to be the -th taxation time point.Thus, , we have

27)
Proof Given that the after-tax excess of the surplus level over u at time u  is exponentially distributed due to the memoryless property of the exponential distribution.By a probabilistic argument, one easily shows that for

When
, the general solution of Equation (3.20) can be expressed as for all 0 u  .

Explicit Results for Erlang(2) Innovation
In ume that i W 's are Erlang(2) dis-

Waiting Times
this section, we ass tributed with parameters 1  and 2  .We also assume u u y g u y u y g u u Applying the operator Copyright © 2013 SciRes.OJS without loss of generality, we assu that me 1 2

 
 .We know that Equation (3.4) ha and 3 r which satisfies s three real roots, say 1 2 , r r Thus, we have 3 3 e ,0 , Apply Equation (2.10) together with Equations (2.3) and (3.5) when  , we get Some calculations give u b   , using the explicit expressions of And, for u b  , we have Then we can get that when i W 's are Erlang (2) distributed with pa

3 ) 1 2 :
For certain distributionsF , one can derive integrodifferential equations for 0   , g u u and   ,b V u .Let us assume that the i.i.d innovation waiting times have a common generalized Erlang  n T S distribution, i.e. the i 's are distributed as the sum of n independent and exponentially distributed r.v.'s The following theorem 2.1 gives the integro-differential equations for   , 0 g u u i when T 's have a generalized Erlang   n I distribution.Theorem 2.1 Let and D denote the identity operator and differentiation operator respectively.Then   0 , u u 0 u b satisfies the following equation for g   Copyright © 2013 SciRes.OJS

. 11 )
Using from Equation (2.10) and from Equation (2.11), we can derive from Equation (2.4) for   Equation (2.6) and Equation (2.7), we have for u b  transform of the first upper exit time  , which is the time until the risk process record high for the first time without leading to ruin   0 0 is the probability that the process b reaches a new record high before ruin.Then the closed-form expression of the quantity g u  can be calculated as follows.
using a simple sample path argument, we immediately have, 25) denotes the -th moment of the accumulated discounted tax payments over the life time of the surplus process 32) Now, it remains to determine the unknown constant C in Equation (3.20).The continuity of   , M u b on b and Equation (3.22) lead to c are arbitrary constants.To dee arbitrary constants, we insert Equatio and (3.6) into Equation (2.3) and obtain in Liu et al.[17], we obtain llow the same steps to get the explicit ex We p that when the innovation times are ntially distributed, one can fo pressions of   g u  , which coincide with the results in Albrecher et al e 3.2 (The expected discounted tax payments.)Following from Equat ve for 0 u b   , .(2008).Exampl ion (2.34) of Theorem 2.2 and Remark 3.1, we ha


15) and (3.17) and the expecte tax payments can be given by Equation(3.20).
It needs to be mentioned that from the proof of Lemma 2.1, we know that