In this paper, we consider the dual risk model in which periodic taxation are paid according to a loss-carry-forward sys tem and dividends are paid under a threshold strategy. We give an analytical approach to derive the expression of g δ ( 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, closed form expressions for the expected discounted tax payments are given.
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 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 [
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
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
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,
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.
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,
When,
It follows from Equation (2.1) and from Equation (2.2) that is continuous on as a function of and that
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
and
for.
Proof First, we rewrite as when
with in the surplus process (1.2)
with. Thus, we have. When,
for, and
By changing variables in from Equation (2.6) and from Equation (2.7), we have for,
for, and
Then, differentiating both sides of from Equation (2.8) and from Equation (2.9) with respect to, one gets
for, and
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
for, and
Again, by changing variables in Equation (2.12) and Equation (2.13) and then differentiating them with respect to, we obtain for
for, and
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
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. [
and
with
In addition, the boundary conditions for are as follows:
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
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
Let and define
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
and when, we have
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
When, the general solution of Equation (3.20) can be expressed as
(2.31)
Due to the facts that and , we have for
Now, it remains to determine the unknown constant C in Equation (3.20). The continuity of on and Equation (3.22) lead to
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
for all.
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
Applying the operator to Equations (2.4) and (2.5) gives
(3.2)
and
(3.3)
The characteristic equation for Equation (3.2) is
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
and
where are arbitrary constants. To determine the arbitrary constants, we insert Equations (3.5) and (3.6) into Equation (2.3) and obtain
and
Apply Equation (2.10) together with Equations (2.3) and (3.5) when, we get
Insert Equation (3.5) into Equation (2.4), we have
In addition, plugging Equations (3.5) and (3.6) into Equation (2.16) yields
and
Some calculations give
with
Remark 3.1 Now, we give the explicit results for
By Equations (3.6) and (3.13), we have for
with
For, using the explicit expressions of in Liu et al. [
with
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,
And, for, we have
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).
The author would like to thank Professor Ruixing Ming and Professor Guiying Fang for their useful discussions and valuable suggestions.