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The optimal reinsurance-investment strategies considering the interests of both the insurer and reinsurer are investigated. The surplus process is assumed to follow a jump-diffusion process and the insurer is permitted to purchase proportional reinsurance from the reinsurer. Applying dynamic programming approach and dual theory, the corresponding Hamilton-Jacobi-Bellman equations are derived and the optimal strategies for exponential utility function are obtained. In addition, several sensitivity analyses and numerical illustrations in the case with exponential claiming distributions are presented to analyze the effects of parameters about the optimal strategies.

The risk management is a significant issue for insurers. Reinsurance and investment are two effective ways to spread risks and gain profits. With reinsurance, the insurer shares part of the risks to the reinsurer and pays reinsurance premium as the compensation. Hence, the quest for optimal reinsurance-investment treaty becomes an attractive topic to both academics and practitioners. Several criteria have been selected to deal with the reinsurance and investment optimization problem. For example, the ruin probability and adjust coefficient are widely used for designing an optimal strategy. Schmidli [

The above-mentioned researches on optimal strategy are from the insurer’s point of view. However, in practice, a reinsurance treaty involves two parties, an insurer and a reinsurer, which have conflicting interests. An optimal reinsurance treaty for an insurer may not be optimal for a reinsurer and it might be unacceptable for a reinsurer as pointed out by Borch [

The aim of present work is to extend parts of works in Liang et al. [

The rest of this paper proceeds as follows. Section

In this section, we formulate a continuous-time model where the insurer and reinsurer are allowed to trade in the insurance market and the financial market with no taxes or fees. Let

Without reinsurance and investment, the surplus process of the insurer is described by the jump-diffusion model [

The insurer is allowed to purchase proportional reinsurance from the reinsurer to transfer the underlying risk. For each

In addition to the reinsurance, the insurer is also allowed to invest in a financial market consisting of one risk-free asset and one risky asset. The price process of the risk-free asset

Let

Suppose that the utility function

In the presence of the proportional reinsurance contract, the surplus process

Let

Suppose that the utility function

Assume that the insurer takes an exponential utility function

For an admissible strategy

Differentiating (

Substituting (

Equation (

Let

Note that

When

When

When

Substituting

Replacing the supremum in (

Here, the stochastic optimal control problem described above has been transformed into solving a partial differential equation for the value function

Let

Following the works in [

Using (

The optimal investment strategy (

From the exponential utility function given by (

The above derivations lead to the following theorem.

Assume that the insurer takes an exponential utility function in the form of (

If

If

If

where

In this section, we discuss the optimal strategy for the reinsurer. Suppose that the utility function of the reinsurer is given by

For an admissible strategy

From Lemma

When

When

When

Substituting

Replacing the supremum in (

Using the approach of Legendre transform, we derive that the dual equation of (

From the exponential utility given by (

The above derivations lead to the following theorem.

Assume that the reinsurer takes an exponential utility function in the form of (

If

If

If

where

A fair reinsurance contract should consider the interests of both insurer and reinsurer, while they both prefer to choose a reinsurance proportion for their own utility maximization. Assume that the insurer and the reinsurer have complete information on the risk of insurance and investment. If the optimal retention level chosen by the insurer is larger than that of the reinsurer, the reinsurer will accept the strategy. But in the opposite case, the reinsurer may oppose the contract proposed by the insurer. When

We give specialized analyses on

In this section, we focus on the exponential distribution of the claim size. Since

In the following two subsections, we analyze the effects of market parameters on the optimal reinsurance-investment strategy and provide several numerical simulations to illustrate our results. Throughout the numerical analyses, unless otherwise stated, the parameters are given by

From (

The effects of

From (

The effects of

Figure

The effects of

The effect of

When

When

From (

The effects of

From Figure

The effects of

In this paper, we investigate two optimization problems which take both the insurer and reinsurer into account. The surplus process follows a jump-diffusion process and the insurer purchases proportional reinsurance from the reinsurer. Two wealth processes are described from the respective view of an insurer and a reinsurer, who are allowed to invest in a risky asset and a risk-free asset. Applying stochastic control theory, we derive the corresponding Hamilton-Jacobi-Bellman equations and obtain the optimal reinsurance-investment strategies for exponential utility maximization. Comparing the optimal strategies designed by two sides, we observe that the insurer’s reinsurance strategy is different from the reinsurer’s strategy. The optimal reinsurance strategies are related to the safety loading of the reinsurer and the claim sizes distribution. If there exists a correlation between the risk model and the price of the risky asset, the optimal investment strategy is influenced by both financial market and insurance market. Otherwise, the investment strategy is only affected by the parameters in the financial market and the risk preference of the investor. When the insurer and the reinsurer have the same absolute reversion risk coefficient, the sum of two optimal proportion retentions equals

The authors declare that there are no conflicts of interest regarding the publication of this paper.