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 [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

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. [17]).

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. [17], we obtain

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.