Optimal Investment under Dual Risk Model and Markov Modulated Financial Market

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.


Introduction
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 λ , which denotes the total number of claims up to time t.Denote the time of arrival of the ith claim by T i and the size of the ith claim by Y i .More details about the surplus process can be found in Asmussen and Albrecher [1], Rolski et al. [2].As pointed out by Albrecher et al. [3], its dual process may also be relevant for companies whose inherent business involves a constant flow of expenses while revenues arrive occasionally due to some contingent events (e.g.discoveries, sales).For instance, pharmaceutical or petroleum companies are prime examples of companies for which it is reasonable to model their surplus process as The past decade has witnessed an increasing attention on the research of dual risk model.For example, see Albrecher et al. [3] for optimal dividend problem, see Cheung and Drekic [4] for dividend approximation and dual risk model with perturbation, see Yao et al. [5] for optimal dividend and equity issuance, see Zhu and Yang [6] for ruin probability under a Markov modulated dual risk model.
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 [7], Browne [8], Fleming and Hernández [9], Hipp and Plum [10], Li et al. [11], Zhang and Siu et al. [12].However, to our best knowledge, there are few papers concentrate on the optimal investment of agent with dual risk process.This is the main contributions of this paper.
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 nonstationary 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. [13].Meanwhile, since Markov-modulated risky model contains several very important stochastic volatility models, thus can be seen as an explanation of many well-known empirical findings, such as the volatility smile, the volatility clustering, and the heavy-tailed nature of return distributions (c.f.Fleming and Hernández [9], Pham [14], Zariphopoulou [15] [16], and references therein).In this paper, the optimal investment problem of an agent with dual risk process under the Markov modulated financial market is studied.By dynamic programming principle, we obtained the HJB equations satisfied by the value function and finished the corresponding verification theorem.A solid example is presented to illustrate how to solve the HJB equation when the claims are exponential distribution.This rest of this paper is organized as follows.In Section 2, the model and problem are introduced.The HJB equation associated with our control problem and the verification theorem for optimal control are investigated in Section 3. In Section 4, we focus on the exponential utility function and closed form expression for optimal investment is obtained.In Section 5, we listed the highlights of this paper and conclusions from the results.

Let ( )
, , Ω   be a complete probability space which carries all random variables to appear in this paper.To proceed our discussion, we introduce the following variables and notations.Let { } , , 1 ,0 , 1 and satisfying the usual conditions.For simplicity, { } , 0 and { } 2 , 0 t B t ≥ are assumed to be mutually independent.
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 ( ) µ ⋅ , ( ) σ ⋅ are the stochastic investment return rate and volatility of the risky market respectively.The dynamic of the external factor is specified by the solution to the following stochastic differential equation (SDE for short) ( ) where [ ] where ( ) r ⋅ is the interest rate function.We interpret the process Z 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 t t X K − is invested into the risk-free asset, then the wealth process of the insurer can be written as following equation.To clarify the impact of investment policy, we adopt as the wealth process of the insurer, then ( ) where 0 x > is the initial surplus of the insurer and c the positive real constant premium rate.Moreover, if at time s T < the wealth of the insurer is x and the external factor is z.Then the wealth process satisfies with the convention that is admissible if it satisfies the following conditions 1) The strategies K t has to be measurable and predictable with respect to the filtration t  ; 2) There is a constant C K which may depend on the strategies K such that P , 0 1.
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 : U →   to be a twice continuously differentiable function, with 0 U ′ > and 0 U ′′ < , then our goal is the following value function: . sup We say that an admissible combined strategies , , , , , .

Properties of Value Function and the Verification Theorem
In this section we embed the problem of maximizing the expected utility from terminal wealth on a finite horizon T < ∞ in the framework of stochastic control theory by dynamic programming method.Then the HJB eq- uation associated with the control problem (2.5) is given by 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 Suppose further that there exist two bounded measurable functions ( ) defines a pair of optimal strategy and , , , , .
where N is the Poisson random measure on , , , , 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 , . .
The proof of the second part of this theorem follows in a similar manner.If we plug K * back into (2.12), for all K ∈ ∆ we obtain By taking the supremum over all K ∈ ∆ in (2.15), we obtain the inequality ( ) ( )   , , , , , , , .
in the last equality, we have .
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.

Existence of a Optimal Pair of Solutions under the Exponential Utility Function
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 ( ) e , 0. 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 ( ) 0 ρ = .Besides Hypothesis 1, we make the following assumptions.
where ( ) , z t ξ will be governed below by a solution to a Cauchy problem.From the definition of ( ) , , e exp e , Plugging these partial derivatives of f into the HJB Equation (3.1), we obtain For simplicity of presentation let us introduce the following notation ( ) Indeed, by a measurable selection theorem, we may find a pair of bounded progressively measurable processes K * satisfied the supremum in (4.8).By substituting K * into (4.8),we obtain the following Cauchy problem: .
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 β is constant, naturally, it is Lipschitz continuous, Hölder continuous, and the operator • Considering Hypothesis 2, we know immediately that ( ) g z is bounded and uniformly Lipschitz conti- nuous.
• Now we show that In fact, by Hypothesis 2, it is clear that the first term of ( ) , d z t is bounded.The second term is bounded by e rT cα .Note that for 0 α > we have ( ) and Hypothesis 2, then it follows from Lemma A.1 that ( ) l z is uniformly Hölder continuous with exponent 1 2 h = , i.e., for all 0 , z z ∈  ( ) ( ) For the second term of ( )  .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 ( ) , z t ξ which satisfies (4.11) and (4.12).The proof of the theorem is now complete.


The aim of the next theorem is to relate the value function V in the form (4.2) to the HJB Equation (3.1) K t In the last line, we used ( ) ≤ for 0 α > and 0 y ≥ .To get condition (3.2), it suffices to obtain an estimate of ( ) In the last inequality, we used Hölder inequality ( , where 1 p < < ∞ and 1 1 1 p q + = ).By considering Hypothesis 2.2 and Theorem A.2, we know that ( ) where C is a positive constant.Moreover, by the Minkovski inequality ( ( ( ) , where 1 p ≤ < ∞ ), one will find that ( ) ( ) ( ) i.e., .
Recall that K is a pair of admissible strategy, by Hölder inequality, we have e exp 4 16 .
, i.e., (3.2) holds for the case 0 r = .In order to prove conditions (3.3) and (3.4), by (4.11) and (4.12) we have e 1 e K t Evidently, (3.3) and (3.4) are easily seen to hold with 0 r = .For the case in which the interest rate 0 r > , let This case is dealt with the same arguments by suitable modification to the first part of the proof.First, we get , exp 2 e 1 exp 2 e .
To accomplish the proof, it is sufficient to prove that ( ) and the fact that ( ) from the first part of the proof, the proof is reduced to showing In fact, by applying (4.15) and with similar arguments to the first part of the proof we have Similarly, since K is a pair of admissible strategy and e t L is a martingale, we also have This completes the proof. 

Highlights and Summary
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 [7] or Li et al. [11]).One should note that when the coefficients are not sensitivity to the changes of the external Markov process, i.e. when the external Markov process changes, the coefficients of the risky market do not oscillate greatly, then our optimal investment policies seem to be very conservative because the optimal investment amount is near to a constant.

Appendix. Parabolic Partial Differential Equations
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.
We say that E is uniformly elliptic, if there is 0 0 , 0 λ λ > such that ( ) Theorem A. 1 (Friedman, 1975).We consider the following Cauchy problem: , in where L is given by ( ) ( ) ( )  Proof.By the mean value theorem and using that ( ) is bounded,we have: , f x f y K x y − ≤ −

1
We say that the strategy bounded first derivative;Considering the form of the utility function, We speculate the following function as a solution to the HJB Equation(3.1)

1 ) 3 ) 1
If the Cauchy problem (A.1) satisfies the following conditions: The coefficients of L are uniformly elliptic;2) The functions , The functions ij a are Hölder continuous in x, uniformly with respect to ( ) then there is a unique solution u of the Cauchy problem (A.1) satisfying Let f be a real positive bounded function with bounded derivative, then f is uniformly Hölder continuous with exponent 1 2 h = , i.e., ( ) ( ) . Model (2.2) covers many Markov modulated risk models, such as the Heston model and a special CIR model.Our model also includes a risk-free asset governed by d z t , combining the mean value theorem and the Hypothesis 2 and Definition 1, Then by the mean value theorem of bivariate functions, we know that there is 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: }