Worst-Case Investment and Reinsurance Optimization for an Insurer under Model Uncertainty

In this paper, we study optimal investment-reinsurance strategies for an insurer who facesmodel uncertainty.The insurer is allowed to acquire new business and invest into a financial market which consists of one risk-free asset and one risky asset whose price process is modeled by a Geometric Brownianmotion.Minimizing the expected quadratic distance of the terminal wealth to a given benchmark under the “worst-case” scenario, we obtain the closed-form expressions of optimal strategies and the corresponding value function by solving the Hamilton-Jacobi-Bellman (HJB) equation. Numerical examples are presented to show the impact of model parameters on the optimal strategies.


Introduction
In recent years, insurance companies are playing the more active role in the financial market.To avoid their risk, insurers are apt to invest in the financial market, purchase reinsurance from the reinsurer and acquire new business (acting as a reinsurer for other insurers).Tremendous literature on optimal investment and/or reinsurance problem for insurers has been investigated.For example, Browne [1] studied the optimal investment problem for an insurer whose surplus is described by a drifted Brownian motion under the criterion of maximizing the expected exponential utility of the terminal wealth or minimizing the ruin probability.To the same question, Hipp and Plum [2] showed a different result when insurance business is modeled by a compound Poisson process.Schmidli [3,4] investigated optimal strategies under the criterion of minimizing the ruin probability.
In addition, it is important to consider model uncertainty due to the uncertainties in the global economy and the debit crisis exiting in financial markets and insurance industries.Meanwhile a growing number of scholars have paid attention to incorporating model uncertainty into optimization problems.For example, Talay and Zheng [14] viewed the Trader and the Market as two players and formulated the model risk control problem as a zero-sum stochastic differential game problem.Mataramvura and Øksendal [15] investigated the risk-minimizing portfolio selection problem in a jump diffusion market.Zhang and Siu [16] studied optimal investment and reinsurance problems via the game theoretic approach and derived closed-form solutions when the objective function is the expected exponential utility of terminal wealth and the expected discounted penalty of ruin, respectively.Lin et al. [17] discussed an optimal portfolio selection problem of an insurer who faces model uncertainty and obtained closed-form solutions to the game problems in both the jump diffusion risk process and its diffusion approximation for the case of an exponential utility by using techniques of stochastic linear-quadratic control.
In this paper, we are concerned about optimal investment and reinsurance strategies for an insurer with model uncertainty.Suppose that the insurer can purchase proportional reinsurance/acquire new business and invest in a simplified continuous-time financial market with a bank account and a risky stock.We measure the risk faced by the insurer by means of the deviation of the surplus to a fixed terminal target (a given benchmark).The insurer's objective is to find an optimal investment-reinsurance strategy under the criterion of minimizing the expected quadratic distance of the terminal wealth to the given benchmark in the "worst-case" scenario.Different from those in Zhang and Siu [16], we propose here a mean-variance portfolio optimization problem under model uncertainty.
The rest of this paper is organized as follows.In Section 2, we give the assumptions and formulate the model dynamics.In Section 3, we derive optimal investment-reinsurance strategy and the corresponding value function with the help of stochastic control theory.In Section 4, we provide some numerical analyses to demonstrate our results.Section 5 concludes the paper.

Model Formulation
Throughout this paper, we work on a filtered complete probability space (Ω, F, P, {F  } 0≤≤ ), where P is a reference probability measure from which a family of real-world probability measures absolutely continuous with respect to P are generated; {F  } 0≤≤ is the filtration generated by all Brownian motions standing for the information available up to time .The finite constant  > 0 is a final time horizon.

Surplus Process.
In what follows, we consider an insurer whose surplus process {(),  ∈ [0, ]} is approximated by a diffusion model.We firstly go back to the classical risk model as follows: Here {  } is the size of the th claim and {  ,  = 1, 2, . ..} are independent and identically distributed (i.i.d.) nonnegative random variables with finite first-order moment  ∞ and second-order moment  2 ∞ ; {  }, independent of {  ,  = 1, 2, . ..}, is a homogeneous Poisson process with intensity ;  > 0 is the premium rate which is assumed to be calculated by the expected value principle, that is,  = (1 + ) ∞ , where  is the safety loading of the insurer and  0 is the initial capital.
According to Grandell [18] or Schmidli [19], the insurer's surplus process can be approximated by the following diffusion model: where  =  ∞ is the premium return rate of the insurer;  2 0 =  2 ∞ represents the volatility of the insurer's surplus; { 0 ()} is a standard Brownian motion.

Financial Market.
The insurance company is allowed to invest the money into a financial market with a riskless asset and a risky asset.Without loss of generality we assume any fractional units of assets can be traded continuously and no taxes or transaction costs are involved in our model.

Wealth Process.
As mentioned before, the insurer is allowed to invest in the financial market and purchase proportional reinsurance or acquire new business as described in Bäuerle [6].We denote by () the money amount invested in the risky asset and by () the retention level of reinsurance.For each  ∈ [0, ], () fl ((), ()) represents the strategy adopted by the insurance company and denote by {  () |  ∈ [0, ]} the surplus of the insurance company with the above notation .Thus the wealth process   () of the insurer can be described as where  =  1 −  0 > 0,  >  is the safety loading of the reinsurance/new business.
A strategy )] < +∞, and ( 5) has a unique strong solution.Denote by A the set of all admissible strategies.
For each  ∈ Θ, a new (real-world) probability measure P  absolutely continuous with respect to P on G() is defined by Then the family P(Θ) of real-world probability measures with index set Θ is generated.

The Basic Problem (MV)
As in Mataramvura and Øksendal [15], we define {() |  ∈ [0, ]}, a vector process, by Suppose the market is the leader of the game and aims to select an admissible strategy {() |  ∈ [0, ]} which represents the "worst-case" scenario of the minimal expected quadratic deviation between the terminal surplus and some preset targets.That leads to the Min-Max problem where  ∈ R + is a certain predefined benchmark.Here it is reasonable to assume  ≥  0   0  , which states that the insurer's terminal wealth  cannot be less than the amount earned by the insurer if all the wealth was invested in the riskfree asset.The market selects an optimal probability scenario { * () |  ∈ [0, ]} to maximize the minimal expected quadratic deviation and the insurer reacts antagonistically by choosing an optimal portfolio processes { * () |  ∈ [0, ]} to minimize the expected quadratic deviation of   () to .
The proof of Theorem 1 is similar to the proof of Theorem 3.2 in Mataramvura and Øksendal [15].So we do not repeat it here.
Combining with Theorem 1, the above analyses are summarized as the following theorem.
Theorem 2. The optimal investment-reinsurance strategy of the Min-Max problem (11) is Moreover, the corresponding value function is given by Remark 3. We find that (i) under model uncertainty the optimal policy  * = ( * ,  * ) depends on the volatility of both the insurer's surplus and the risky asset; (ii) if we do not take account of the model risk, the corresponding optimal investment-reinsurance strategies (see Appendix) are different: the volatility of the insurer's surplus has no influence on the investment strategy π() (see Appendix (A.11)), and the volatility of the risky asset has no influence on the reinsurance strategy ã() (see Appendix (A.12)).

Numerical Results
In this section, we present some numerical illustrations and sensitivity analysis of optimal investment-reinsurance strategies.Throughout the numerical analysis, unless otherwise stated, the basic parameters are given by  0 = 10,  = 1,  1 = 1,  = 1.5,  = 1.1,  0 = 0.04,  1 = 0.2,  0 = 1.5,  1 = 1.15,  = 15,  = 5, and  = 300.In the following numerical examples, we illustrate the effect of different parameters on the optimal strategies by varying the value of one parameter with others fixed each time.
Figure 1 shows the effect of the riskless rate of interest  0 , the appreciation rate of the risky asset  1 , the volatility of the insurer's surplus  0 , and the volatility of the risky asset  1 on the optimal investment strategy  * ().We compare the investment strategies in Figure 1(a) when  0 = 0.01 and  0 = 0.06 for fixed  ∈ [0, 5].It is shown that as the riskfree interest rate  0 increases, the optimal investment strategy  * () moves up in any time.And when  is near time 0, the trend is very obvious.From Figures 1(b) and 1(c), we find that  * () increases with respect to  1 and  0 for fixed  ∈ [0, 5] where we compare the investment strategies when  1 = 0.05 and  1 = 0.35 and  0 = 1.15 and  0 = 1.5.It is a different case as in Figure 1(d), where we compare the investment strategies when  1 = 0.55 and  1 = 1.65 for fixed  ∈ [0, 5].
Figure 2 shows the effect of different parameters on the optimal reinsurance strategy.In Figure 2(a), we compare the reinsurance strategies when  0 = 0.01 and  0 = 0.06 for fixed  ∈ [0, 5].It is shown that as the risk-free interest rate  0 increases, the optimal investment strategy  * () moves down.In Figure 2 We can find from the above two groups of Figures that, while keeping other parameters unchanged: (i) The insurer is inclined to invest in the market (compared with buying reinsurance) when the riskless rate of interest  0 increases.
(ii) When the appreciation rate of the risky asset  1 increases, the insurer is inclined to invest in the market (compared with buying reinsurance) during the first half of the horizon and to buy reinsurance (compared with investing in the market) during the second half of the horizon.
(iii) As the volatility of the insurers surplus  0 increases, the insurer seems to be willing to invest in the market during the first half of the horizon and buy reinsurance during the second half of the horizon.
(iv) The insurer is inclined to invest in the market (compared with buying reinsurance) when the volatility of the risky asset  1 increases.

Conclusion
In this paper, we study an optimization problem for an insurer when facing uncertainties.The insurer is allowed to invest into a financial market and purchase proportional reinsurance/acquire new business.The model uncertainty is described by a family of probability measures equivalent to the original probability measure.Compared with Zhang and Siu [16] and Lin et al. [17], we focus on the criterion of minimizing the expected quadratic distance of the insurer's terminal wealth to a given benchmark.Under the "worstcase" scenario, we have derived the optimal investmentreinsurance strategy and the corresponding value function explicitly.In addition, we have presented some numerical illustrations and sensitivity analysis to show the effect of parameters on optimal strategies.(A.9) (b), we compare the reinsurance strategies when  1 = 0.05 and  1 = 0.35 for fixed  ∈ [0, 5].In Figure 2(c), we compare the reinsurance strategies when  0 = 1.15 and  0 = 1.55 for fixed  ∈ [0, 5].In Figure 2(d), we compare the reinsurance strategies when  1 = 0.55 and  1 = 1.65 for fixed  ∈ [0, 5].
Effect of  1 on investment strategy

Figure 1 :
Figure1: The effect of parameters on optimal investment strategy.
Effect of  1 on reinsurance strategy