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 , 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.
 Bai, L. and Guo, J. (2008) Optimal Proportional Reinsurance and Investment with Multiple Risky Assets and No-Shorting Constraint. Insurance: Mathematics and Economics, 42, 968-975.
 Chen, S., Li, Z. and Li, K. (2010) Optimal Investment-Reinsurance Policy for an Insurance Company with VaR Constraint. Insurance: Mathematics and Economics, 47, 13-24.
 Bai, L. and Zhang, H. (2008) Dynamic Mean-Variance Problem with Constrained Risk Control for the Insurers. Mathematical Methods of Operations Research, 68, 181-205.
 Zeng, Y. and Li, Z. (2011) Optimal Time-Consistent Investment and Reinsurance Policies for Mean-Variance Insurers. Insurance: Mathematics and Economics, 49, 145-154.
 Xu, L., Wang, R. and Yao, D. (2008) On Maximizing the Expected Terminal Utility by Investment and Reinsurance. Journal of Industrial and Management Optimization, 4, 801-815.
 Yi, B., Li, Z., Viens, F.G. and Zeng, Y. (2013) Robust Optimal Control for an Insurer with Reinsurance and Investment under Hestons Stochastic Volatility Model. Insurance: Mathematics and Economics, 53, 601-614.
 Li, Z., Zeng, Y. and Lai, Y. (2013) Optimal Time-Consistent Investment and Reinsurance Strategies for Insurers under Hestons SV Model. Insurance: Mathematics and Economics, 51, 191-203.
 Sun, Z., Zheng, X. and Zhang, X. (2017) Robust Optimal Investment and Reinsurance of an Insurer under Variance Premium Principle and Default Risk. Journal of Mathematical Analysis Applications, 446, 1666-1686.
 Gu, A., Viens, F.G. and Yi, B. (2017) Optimal Reinsurance and Investment Strategies for Insurers with Mispricing and Model Ambiguity. Insurance: Mathematics and Economics, 72, 235-249.
 Zeng, Y., Li, D., Chen, Z. and Yang, Z. (2018) Ambiguity Aversion and Optimal Derivative-Based Pension Investment with Stochastic Income and Volatility. Journal of Economic Dynamics and Control, 88, 70-103.
 Gu, A., Viens, F.G. and Yao, H. (2018) Optimal Robust Reinsurance-Investment Strategies for Insurers with Mean Reversion and Mispricing. Insurance: Mathematics and Economics, 80, 93-109.