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# Optimal reinsurance and investment strategies for an insurer and a reinsurer under Hestons SV model: HARA utility and Legendre transform

• * Corresponding author: Peibiao Zhao

This work was supported by NNSF of China (No.11871275; No.11371194)

• The present paper investigates an optimal reinsurance-investment problem with Hyperbolic Absolute Risk Aversion (HARA) utility. The paper is distinguished from other literature by taking into account the interests of both an insurer and a reinsurer. The insurer is allowed to purchase reinsurance from the reinsurer. Both the insurer and the reinsurer are assumed to invest in one risk-free asset and one risky asset whose price follows Heston's SV model. Our aim is to seek optimal investment-reinsurance strategies to maximize the expected HARA utility of the insurer's and the reinsurer's terminal wealth. In the utility theory, HARA utility consists of power utility, exponential utility and logarithmic utility as special cases. In addition, HARA utility is seldom studied in the optimal investment and reinsurance problem due to its sophisticated expression. In this paper, we choose HARA utility as the risky preference of the insurer. Due to the complexity of the structure of the solution to the original Hamilton-Jacobi-Bellman (HJB) equation, we use Legendre transform to change the original non-linear HJB equation into its linear dual one, whose solution is easy to conjecture in the case of HARA utility. By calculations and deductions, we obtain the closed-form solutions of optimal investment-reinsurance strategies. Moreover, some special cases are also discussed in detail. Finally, some numerical examples are presented to illustrate the impacts of our model parameters (e.g., interest and volatility) on the optimal reinsurance-investment strategies.

Mathematics Subject Classification: Primary: 91B30, 93E20; Secondary: 62P05.

 Citation: • • Figure 1.  Effect of $x$ on $q^*_{HARA}$

Figure 2.  Effects of $y$ on pH ARA*

Figure 3.  Effect of $r$ on $q^*_{\exp}$ and $p^*_{\exp}$

Figure 4.  Effect of v on qexp* and pexp*

Figure 5.  Effect of r on qexp* and pexp*

Figure 6.  Effect of x on π1 HARA*

Figure 7.  Effect of y on π2HARA*

Figure 8.  Effect of v on π1HARA* and π2HARA*

Figure 9.  Effect of β on π1 exp*

Figure 10.  Effect of α on π1 exp*

Figure 11.  Effect of σ on π1 exp*

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