Reinsurance and investment are the main tools for insurers to manage the risk and profit. Insurers can transfer the risk to reinsurers by purchasing reinsurance; meanwhile, they would invest their surplus into financial market to pursue extra profit. Recently, the problem of optimal reinsurance and investment has attracted great interest. For example, Schmidli  , Bai and Guo  and Chen et al.  investigated the optimal investment-reinsurance strategies for insurers to minimize the ruin probability; Bäuerle  , Bai and Zhang  , Zeng and Li  studied the optimal reinsurance and investment strategies for insurers with mean-variance criteria; Maximizing the expected utility from terminal wealth was investigated by many literature, see among Yang and Zhang  , Wang  , Xu et al.  and so on.
Theoretically, real markets are full of friction. Hence, there exists mispricing between a pair of assets. We can find vivid examples of mispricing in certain Chinese companies (such as Bank of China) traded on both Chinese stock exchanges as share A and Hong Kong stock exchanges as shares. Yi et al.  first consider a dynamic portfolio problem with mispricing and model ambiguity. Gu et al.  discuss optimal proportional reinsurance-investment problem for an insurer with mispricing and model ambiguity. In this paper, we seek the optimal proportional reinsurance-investment problem with mispricing under observed mean-reverting stochastic risk premium.
Under the criteria of maximizing the expected utility of terminal wealth, most of the literature mentioned above are based on expected value premium principle due to its simplicity and popularity in practice. Sun et al.  consider the optimal investment-reinsurance strategies under variance premium principle. Zeng et al.  and Gu et al.  investigate the optimal proportional reinsurance-investment problem for an insurer with mispricing, model ambiguity with mean-reversion under expected value premium principle. Motivated by these papers, both the insurance and reinsurance premium payments are calculated by using the variance premium principle in this paper.
The rest of the paper is organized as follows. In Section 2, we provide the financial market and the insurance model. In Section 3, the optimal robust proportional reinsurance-investment problem is established. Optimal proportional reinsurance-investment strategies and their corresponding optimal value functions are given in Section 4.
2. Economy and Assumption
In this section, we formulate a continuous-time financial model where the insurers can trade in the financial market and in the insurance market with no taxes or fees. Let be a probability space, in which is the state space and is a σ-algebra on . is a fixed constant, representing the time horizon, is a filtration, which describes the flow of information over time, the σ-algebra describes the information available up to time t, and satisfies the usual condition (it contains all P-null sets and is right continuous). We denote P as a reference measure and suppose that all stochastic processes given in the following are assumed to be adapted on this space.
2.1. Surplus Process
We assume that the insurer’s surplus is given by the classical Cramér-Lunderberg risk model (without reinsurance and investment), the insurer’s surplus R is given by
where c is the premium rate, the claim arrival process is a Poisson process with constant intensity and the random variables are i.i.d claim sizes independent of . We let denote the claim size distribution with finite first-order moment and second-order moment . The stochastic process is a standard Brownian motion independent of N, representing the diffusion risk of the surplus process. The premium rate c is assumed to be calculated via the expected value principle, i.e.,
where is the insurer’s safety loading.
In this paper, the insurer can purchase proportional reinsurance or acquire new business to adjust the exposure to insurance risk. The proportional reinsurance/new business level is associated with the risk exposure at time t. When , it means the insurer purchases proportional reinsurance. In this case, for each claim, the insurer only pays its , while the reinsurer pays the rest for each claim. When , it means the insurer acquires new business from the other insurers as a reinsurer. Then, the aggregate reinsurance premium under the variance principle takes the form
with . Thus, the surplus process of the insurer after taking into account reinsurance is governed by
where defined on is a Poisson random measures used to represent the compound Poisson process as
If we denote , then
and is the compensator of the random measure . Thus, the compensated measures is related to the compound Poisson process as follows
2.2. Financial Market
We assume that financial market consists of one risk-free asset, and a pair of stocks with mispricing (two price processes for the same asset on two distinct stock markets). We assume for the moment that the price of risk-free asset is given by
where constant is the risk-free interest rate. The price process of the market index is expressed as
where the market risk premium , the market volatility are positive constants, and is a standard Brownian motion on . The two mispriced price processes are modeled as a pair of stocks and which are coupled via the pricing error
We assume the vector solves the system of stochastic differential equations
in which are constant parameters. And we assume the premium for the common risk part is the observed mean-reverting process whose dynamics are given by
where is the long-run mean of the risk premium, and is the degree of mean reversion (or mean-reversion rate). The term shows the effect of mispricing on the ith stock’s price. The term describe the idiosyncratic risk of stock i, describes the common risk, while the individual risk is generated by the asset i, . The term shows the effect of miscricing on ith stock’s price via the pricing error defined above. We assume that all standard Brownian motions , , and are independent of each other and all are independent of . Based on (11) and (12), ordinary stochastic calculus implies that the dynamics of the pricing error is given by the following equation
Thus Equation (14) shows that X is also a MR process. Note that is the correlation coefficient between the Brownian motions and , and is the correlation coefficient between and . Moreover, there exist Brownian motions and satisfying the following equations:, where. Moreover, we assume all standard Brownian motions , , , , and are independent each other and all are independent of . To capture the features of mispricing, we assume .
In addition to reinsurance, the insurer can invest in our financial market. Thus we denote by the corresponding investment strategy, where shows the amount of the wealth invested in the market index, , and denote the amounts invested in the two stocks, respectively, hence the remainder, is invested in the risk-free asset.
2.3. Wealth Process
We denote the whole reinsurance-investment strategy by . As a result of adopting the strategy , the insurer’s corresponding reserve satisfies the following stochastic dynamics:
where and is the insurer’s initial wealth.
3. Robust Problem with Mispricing
To derive explicit results, we need to make some assumptions regarding the ambiguity-averse insurer’s utility. Suppose that the insurer has exponential utility function, i.e.
where is a constant representing the coefficient of absolute risk aversion. The insurer is assumed to be ambiguity-neutral with objective function
where is the set of admissible strategies for an ambiguity-neutral insurer in a given market and represents the conditional expectation under the probability measure P.
Definition 3.1. A strategy is said to be admissible, if
1) , ;
2) is predictable with respect to and , where ;
3) , Equation (5) has a unique solution with , where is the chosen model to describe the worst case and .
Denote by the set of all admissible strategies.
In order to incorporate the ambiguity, we assume that the insurer does not have full confidence in the reference probability P. She uses the probability P as her reference probability for the wealth process, but takes some alternative models at the same time. Every alternative model is characterized by another stochastic process and the associated probability measure Q, which is equivalent to the reference measure P. We denote this class of probability measures by : .
The change of measure from P to Q can be defined by its Randon-Nikodym derivative, i.e., there exists a progressively measurable process , such that every Q in satisfies
where , and . According to Girsanov’s theorem (Æksendal  ), under the alternative measure Q, the stochastic process , , , , and are standard Brownian motions, where
Moreover, the intensity of the Poisson process becomes , that is,
is a martingale. For tractability and ease of interpretation, the distribution of the claim Y is assumed to be known, and is restricted to be identical under P and Q. Thus, the dynamics of the wealth process under Q is
Meanwhile, the dynamics of the mispricing error under Q can be given by
and (13) can be modified to
Following from Maenhout  , Gu et al.  and Zeng et al.  , we show that the increase in relative entropy from t to equals
where , ,
. For convenience, similar to Maenhout  and Gu et al.  , we assume that , and are non-negative and state-dependent functions which are inversely proportional to the value function:
where , and represent the insurer’s ambiguity aversion levels to the diffusion modeling and jump modeling risk.
Next we aim to derive the explicit solution to the HJB Equation (22) with preference parameter (27).
4. Optimal Robust Investment and Reinsurance Strategy
The purpose of this section is to find the optimal investment strategy and the optimal proportional reinsurance strategy under the worst-case scenario. According to the principle of dynamic programming, the robust Hamilton-Jacobi-Bellmann (HJB) equation established by Anderson et al.  to express the value function (26), can be derived as
with the boundary condition , and , , , , , , and represent the value function’s partial derivative w.r.t the corresponding variables.
In order to obtain the solution J of (28), we conjecture that has the ansatz form
with the boundary condition , . A direct calculation yields partial derivatives
Substituting (30) back into Equation (28) and according to the first-order conditions for , we can obtain the minimum point given by
We substitute (31) into the HJB Equation (28), and differentiating w.r.t. implies
Differentiating Equation (32) w.r.t. , we get the optimal reinsurance strategy satisfies
Using the result of Gu et al.  , we get . If this were not true, then and . This would cause a contradiction with (33).
According to the first-order condition for , and , we have
Inserting , , , into (32) and letting the coefficients of w, a, x, ax, and be zero, we get
Taking into account the boundary conditions , , we know that
where , . Using the verification of Yi et al.  , we have the following theorem.
Theorem 4.1. For the optimization problem (26), the value function is given by
where , , , , , and are given by (38)-(43), and the corresponding optimal reinsurance-investment strategy is given by
where is given by Equation (33), and , , are given by Equations (34)-(36).
If all the ambiguity-aversion coefficients equal 0, i.e. in our model, the optimization problem degenerates into optimization problem (17) without ambiguity aversion. Similarly to Theorem 4.1, we have the following Proposition.
Proposition 4.1. For optimization problem (17), the value function is given by
and the optimal strategy , where , , , , , and are given by (46)-(52)
where , .
If the insurer only invests in the financial markets, and does not purchase reinsurance, the optimization problem (26) degenerates to investment only problem. With in Theorem 4.1, we easily derive the following proposition.
Proposition 4.2. For optimization problem (26), if only investment is discussed, the value function is given by
where , , , and are given by Theorem 4.1, and
The research was supported by the National Natural Science Foundation of China (No.11501319), the Education Department of Shandong Province Science and Technology Plan Project (No. J15L105), the Natural Science Foundation of Shandong Province (No. ZR2015AL013) and the China Postdoctoral Science Foundation (No. 2015M582064). The author is a doctoral researcher in center for post-doctoral studies of Qufu Normal University.
The authors declare that they have no conflict of interests.
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