The Expected Discounted Tax Payments on Dual Risk Model under a Dividend Threshold ()
1. Introduction
Consider the surplus process of an insurance portfolio
(1.1)
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 
have common distribution
. We also assume that the innovation sizes
, independent of
, 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 company 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 
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 
can be expressed as
(1.2)
for
, with
. where 
is the indicator function of event 
and 
is the surplus immediately before time
.
For practical consideration, we assume that the positive safety loading condition
(1.3)
holds all through this paper. The time of ruin is defined as 
with 
if
for all
.
For initial surplus
, let 
be the present value of all dividends until ruin, and 
is the discount factor. Denote by 
the expectation of
, that is,
(1.4)
It needs to be mentioned that we shall drop the subscript 
whenever 
is zero.
The rest of this paper is organized as follows. In Section 2, We derive the expression of 
(i.e. 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.
2. Main Results and Proofs
Let 
denote the Laplace transform of the upper exit time
, which is the time until the risk process 
starting with initial capital 
up-crosses the level 
for the first time without leading to ruin before that event. In particular, 
is the probability that the process 
up-crosses the level 
before ruin.
For general innovation waiting times distribution, one can derive the integral equations for
. When
,
(2.1)
When
,
(2.2)
It follows from Equation (2.1) and from Equation (2.2) that 
is continuous on 
as a function of 
and that
(2.3)
For certain distributions
, one can derive integrodifferential equations for 
and
. Let us assume that the i.i.d innovation waiting times have a common generalized Erlang
distribution, i.e. the
’s are distributed as the sum of n independent and exponentially distributed r.v.’s 
with 
having exponential parameters
.
The following theorem 2.1 gives the integro-differential equations for 
when
’s have a generalized Erlang
distribution.
Theorem 2.1 Let 
and 
denote the identity operator and differentiation operator respectively. Then 
satisfies the following equation for 
(2.4)
and
(2.5)
for
.
Proof First, we rewrite 
as 
when
with 
in the surplus process (1.2)
with
. Thus, we have
. When
,
(2.6)
for
, and
(2.7)
By changing variables in from Equation (2.6) and from Equation (2.7), we have for
,
(2.8)
for
, and
(2.9)
Then, differentiating both sides of from Equation (2.8) and from Equation (2.9) with respect to
, one gets
(2.10)
for
, and
(2.11)
Using from Equation (2.10) and from Equation (2.11), we can derive from Equation (2.4) for 
on
.
Similar to from Equation (2.6) and Equation (2.7), we have for 
(2.12)
for
, and
(2.13)
Again, by changing variables in Equation (2.12) and Equation (2.13) and then differentiating them with respect to
, we obtain for 
(2.14)
for
, and
(2.15)
Using Equation (2.14) and Equation (2.15), we obtain Equation (2.5) for 
on
.□
It needs to be mentioned that from the proof of Lemma 2.1, we know that
Therefore, Equations (2.10), (2.11), (2.14) and (2.15) yield
(2.16)
Remark 2.1 Using a similar argument to the one used in Lemma 2.1, one can get that when the innovation waiting times follow a common generalized Erlang
distribution, the expected discounted dividend payments 
satisfies the following integro-differential equation (see Liu et al. [17]).
(2.17)
and
(2.18)
with
(2.19)
In addition, the boundary conditions for 
are as follows:
(2.20)
(2.21)
with Equation (2.19).
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 process
. We claim that the process
shall up-cross the initial surplus level 
at least once to avoid ruin.
Now, let
(2.22)
denote the Laplace transform of the first upper exit time
, which is the time until the risk process
starting with initial capital 
reaches a new record high for the first time without leading to ruin before that event. In particular, 
is the probability that the process 
reaches a new record high before ruin. Then the closed-form expression of the quantity 
can be calculated as follows.
When
. When
, using a simple sample path argument, we immediately have,
or, equivalently
(2.23)
Let 
and define
(2.24)
to be the 
-th taxation time point. Thus,
(2.25)
denotes the 
-th moment of the accumulated discounted tax payments over the life time of the surplus process
.
We will consider a recursive formula of 
in the following theorem 2.2.
Theorem 2.2 When
, we have
(2.26)
and when
, we have
(2.27)
Proof Given that the after-tax excess of the surplus level over 
at time 
is exponentially distributed with mean 
due to the memoryless property of the exponential distribution. By a probabilistic argument, one easily shows that for 
(2.28)
Differentiating with respect to 
yields
(2.29)
When
, we have
(2.30)
When
, the general solution of Equation (3.20) can be expressed as
(2.31)
Due to the facts that 
and 
, we have for 
(2.32)
Now, it remains to determine the unknown constant C in Equation (3.20). The continuity of 
on 
and Equation (3.22) lead to
(2.33)
Plugging Equation (2.33) into Equation (2.30), we arrive at Equation (2.26). □
The special case 
leads to an expression for the expected discounted total sum of tax payments over the life time of the risk process
(2.34)
for all
.
3. Explicit Results for Erlang(2) Innovation Waiting Times
In this section, we assume that
’s are Erlang(2) distributed with parameters 
and
. We also assume that 
(without loss of generality).
Example 3.1 Note that
(3.1)
Applying the operator 
to Equations (2.4) and (2.5) gives
(3.2)
and
(3.3)
The characteristic equation for Equation (3.2) is
(3.4)
without loss of generality, we assume that
. We know that Equation (3.4) has three real roots, say 
and 
which satisfies
With 
replace 
in Equation (3.4), we get the characteristic equation of Equation (3.3), whose roots are denoted by 
and 
with
Thus, we have
(3.5)
and
(3.6)
where 
are arbitrary constants. To determine the arbitrary constants, we insert Equations (3.5) and (3.6) into Equation (2.3) and obtain
(3.7)
and
(3.8)
Apply Equation (2.10) together with Equations (2.3) and (3.5) when
, we get
(3.9)
Insert Equation (3.5) into Equation (2.4), we have
(3.10)
In addition, plugging Equations (3.5) and (3.6) into Equation (2.16) yields
(3.11)
and
(3.12)
Some calculations give
(3.13)
with
(3.14)
Remark 3.1 Now, we give the explicit results for
By Equations (3.6) and (3.13), we have for 
(3.15)
with
(3.16)
For
, using the explicit expressions of 
in Liu et al. [17], we obtain
(3.17)
with
(3.18)
where
and
We point out that when the innovation times are exponentially distributed, one can follow the same steps to get the explicit expressions of
, which coincide with the results in Albrecher et al. (2008).
Example 3.2 (The expected discounted tax payments.) Following from Equation (2.34) of Theorem 2.2 and Remark 3.1, we have for
,
(3.19)
And, for
, we have
(3.20)
Then we can get that when
’s are Erlang (2) distributed with parameters 
and
, the expresses of 
can be given by Equations (3.15) and (3.17) and the expected discounted tax payments can be given by Equation (3.20).
4. Acknowledgements
The author would like to thank Professor Ruixing Ming and Professor Guiying Fang for their useful discussions and valuable suggestions.
NOTES
#Corresponding author.