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In this paper, the optimal investment problem for an agent with dual risk model is studied. The financial market is assumed to be a diffusion process with the coefficients modulated by an external process, which is specified by the solution to a kind of stochastic differential equation. The object of the agent is to maximize the expected utility from terminal wealth. Together with the regularity property of the value function, by dynamic programming principle, the value function of our control problem is turned to be the unique solution to the associated Hamilton-Jacob-Bellman (HJB for short) equation. When the utility is an exponential function with constant risk aversion, close form expressions for value function and optimal investment policy are obtained.

The classical surplus process of an insurer is given by

where x > 0 is the initial surplus, c is the positive constant premium income rate, N_{t} is Poisson process with parameter_{i} and the size of the ith claim by Y_{i}. More details about the surplus process can be found in Asmussen and Albrecher [

The past decade has witnessed an increasing attention on the research of dual risk model. For example, see Albrecher et al. [

As we all know, investment is an important element in the financial agent for which can bring them potential profit. Thus, optimal investment for insurers has drawn great attentions in recent years, for example, see the works of Bai and Guo [

Usually, the coefficients of the dynamics of the financial market are assumed to be constant. However, in reality, the returns from the risky assets might not be constants. So, it would be of practical relevance and importance to consider asset pricing models with non-constant coefficients, which can incorporate the feature of non- stationary returns. Among all kinds of stochastic coefficients models, Markov-modulated risky model has been recognized recently as an important feature to asset price models. There is much literature documenting such models in assets returns, such as French et al. [

Let

and satisfying the usual conditions. For simplicity,

Assume that there are two kinds of asset available for investors, one risky asset and one risk free asset. The risky asset is assumed to be

where

where

where _{t} as the behavior of some economic factor that has an impact on the dynamics of the risky asset and the risk-free asset price. In this paper, we allow the company takes an investment strategy into account when making decisions. Then if X_{t} is the company’s wealth, and let K_{t} denote the amount invested into the risky asset at time t. The remained reserve

where

with the convention that

Definition 1 We say that the strategy

1) The strategies K_{t} has to be measurable and predictable with respect to the filtration

2) There is a constant C_{K} which may depend on the strategies K such that

We denote the set of admissible strategies as

Suppose that the company is interested in maximizing the expected utility of wealth at time T. Without loss of generality, we can define the utility function

We say that an admissible combined strategies

Hypothesis 1 1) The functions

2) The function

In this section we embed the problem of maximizing the expected utility from terminal wealth on a finite horizon

with terminal condition

The following verification theorem shows that under some proper conditions, a solution to previous HJB equation provides us the optimal investment policy.

Theorem 2 (The Verification Theorem) Suppose that there is a smooth solution

Then for each

Suppose further that there exist two bounded measurable functions

then

Proof. Let

where

Compensate (3.5) by

we have

Assumptions (3.3) and (3.4) mean that

are martingales. Assumption (2.8) implies that

is a martingale. Then, by taking the expectation on both sides of (2.11) yields that

Note that f is a smooth solution to HJB equation (2.6), we have

That is to say for any

Taking

The proof of the second part of this theorem follows in a similar manner. If we plug

By taking the supremum over all

By considering (2.13) for all

Letting

Moreover, by recalling (2.14), it is easy to find that

This completes the proof.

Remark 1 Classical method of applying HJB equation for solving optimal control problems is pre-assume (or find) that there exist a smooth solution to the HJB equation, and then finish the argument by verification theorem. However, the HJB equations do not always admit classical solution, and thus the verification theorem invalid. In this case, viscosity solution will be introduced to cover the connections between the optimal control problem and the HJB equation. However, in next section, we exploit a closed representation of the solution to the HJB Equation (3.1) when the utility function is an exponential type. By this results, we further find the closed from optimal investment policy and the expressions of value function. As to very general utility function, it is difficult to find closed form solutions to HJB equation and we leave it as future research.

In this section, we devote to the existence and uniqueness of the solution of the HJB Equation (3.1) when the preferences of the insurer are exponential, i.e., the utility function is governed by

In order to get a linear PDE, in the remainder of this paper we consider only the case where the correlation coefficient is equal to zero

Hypothesis 2 1)

2) g is uniformly Lipschitz and bounded;

3)

Considering the form of the utility function, We speculate the following function as a solution to the HJB Equation (3.1)

where

Plugging these partial derivatives of f into the HJB Equation (3.1), we obtain

For simplicity of presentation let us introduce the following notation

It is trivial to see that the supremum is achieved at

Indeed, by a measurable selection theorem, we may find a pair of bounded progressively measurable processes

The following theorem asserts the existence and uniqueness of aforementioned Cauchy equation (4.9).

Theorem 3 Assume that

Then the Cauchy problem given by (4.9) has a unique solution, which satisfies the following conditions:

where C_{1} and C_{2} are constants.

Proof. The theorem will be proved if we can show that the Cauchy problem given by (4.9) satisfies the conditions of the Theorem A.1. So we just need to check them.

・ Since

・ Considering Hypothesis 2, we know immediately that

・ Now we show that

is bounded and uniformly Hölder continuous in compact subsets of

In fact, by Hypothesis 2, it is clear that the first term of

Thus

Noting that

For the second term of

Then

Then by the mean value theorem of bivariate functions, we know that there is

where

By (4.10), we obtain that

Since the Cauchy problem (4.9) is homogeneous with a constant terminal condition, then the right-hand side of (4.9) satisfies the property of linear growth and continuous. Finally, the conditions of Theorem A.1, it is easy to find that the Cauchy problem (4.9) has a unique solution

The aim of the next theorem is to relate the value function V in the form (4.2) to the HJB Equation (3.1) in the form of the Cauchy problem (4.9).

Theorem 4 If (4.10) are satisfied, then the value function defined by (2.5) has the form:

where

then the investment strategy

and

Proof. We have already verified that

is a smooth solution of the HJB Equation (3.1). To prove that

Firstly, we consider the case in which

In the last line, we used

We find that

In the last inequality, we used Hölder inequality (

where C is a positive constant. Moreover, by the Minkovski inequality

(

i.e.,

One should note that

and

Recall that K is a pair of admissible strategy, by Hölder inequality, we have

Since

This indicates that

and

Evidently, (3.3) and (3.4) are easily seen to hold with

By

This case is dealt with the same arguments by suitable modification to the first part of the proof. First, we get

and

To accomplish the proof, it is sufficient to prove that

Note that

and the fact that

Similarly, since K is a pair of admissible strategy and

This completes the proof.

The main contributions of this paper include:

・ Both stochastic coefficients financial model and dual risk model are taken into account.

・ Rigorous proof of verification theorem for optimal policy is provided and closed form expressions for optimal policies and value function are derived.

・ A solid example is presented to illustrate how to solve the HJB equation.

As a result, we find that the optimal investment policy is a function of the state of the external Modulate Markov process. When there is no modulated process, the model considered in this paper is reduced to the optimal investment problem under the risky market with stationary coefficient and our results cover those existing results (see Bai and Guo [

The authors are very grateful to anonymous referees’ detailed comments and suggestions, which makes this paper much better. Lin Xu would like to acknowledge the support of the National Natural Science Foundation of China (Grant No. 11201006). Zhu Dongjin would like to acknowledge the support of Major Projects of Colleges and Universities in Anhui Province Natural Science Foundation (KJ2012ZD01).

To illuminate the expression of our research problem, now we introduce and summarize some important results on parabolic PDEs, which play a key role in the proof of Theorem 4.1 (existence and uniqueness theorem), and some terminology and definitions are introduced. We believe that this work will be useful in the development of this paper.

Definition 5 Let

1) We say that E is uniformly elliptic, if there is

for all

2) A function f on

3) f is said to be uniformly Hölder continuous in

Theorem A.1 (Friedman, 1975). We consider the following Cauchy problem:

where L is given by

If the Cauchy problem (A.1) satisfies the following conditions:

1) The coefficients of L are uniformly elliptic;

2) The functions

3) The functions

4) The function

5) The function

6) The function

Lemma A.1 Let f be a real positive bounded function with bounded derivative, then f is uniformly Hölder continuous with exponent

Proof. By the mean value theorem and using that

where K is a constant. By f is positive,we have

i.e.

The proof of this Lemma is now complete.

Theorem A.2 (Pham, 1998). Let

with a standard Brownian motion B_{s}. We assume that for some

for all