Optimal reinsurance-investment problem with dependent risks based on Legendre transform

Zhang, Yan; Zhao, Peibiao

doi:10.3934/jimo.2019011

Cited by 11 publications

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“…(β+σα) 2 and k = 0, considering terminal conditions A 1 (T ) = 0 and A 2 (T ) = 0 by the method of [33], we arrive at…”

Section: Definition 31 Letmentioning

confidence: 99%

“…Based on Legendre transform, we can obtain the explicit expression for the optimal reinsurance-investment strategy with power utility, which is seldom studied in the reinsurance-investment problems. Besides, our method can be used to solve the optimal problem under Vasicek Model in [25] as well as that under CEV Model in [33].…”

Section: Definition 31 Letmentioning

confidence: 99%

“…[5] derived the optimal consumption-investment strategy under HARA utility. [33] introduced HARA utility into the reinsuranceinvestment problem, and investigated an optimal reinsurance-investment problem in relation to thinning dependent risks based on Legendre transform.…”

mentioning

confidence: 99%

“…However, the HJB equation in this paper is a non-linear second order partial differential equation (PDE), it is difficult to simplify or solve this equation directly in general. Motived by [5] and [33], we adopt the Legendre transform-dual technique, instead of ordinary method used in the existing literature, to change the HJB equation into its dual one, the form of whose solution is easy to be conjectured. Applying variable change technique, we achieve the closed-form expression of optimal investment and reinsurance strategies for HARA utility.…”

mentioning

confidence: 99%

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

Zhang¹,

Zhao²,

Teng³

et al. 2021

JIMO

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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.2010 Mathematics Subject Classification. Primary: 91B30, 93E20; Secondary: 62P05.

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“…(β+σα) 2 and k = 0, considering terminal conditions A 1 (T ) = 0 and A 2 (T ) = 0 by the method of [33], we arrive at…”

Section: Definition 31 Letmentioning

confidence: 99%

Section: Definition 31 Letmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

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

Zhang¹,

Zhao²,

Teng³

et al. 2021

JIMO

Self Cite

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“…Yuen et al [27], Liang and Yuen [18], Han et al [15], where only the optimal proportional reinsurance problem is addressed, and see e.g. Bi et al [2,3], Zhang and Zhao [28] which also include the optimal investment problem. To the best of our knowledge, there are only a few papers, where the modeling framework allows for a common shock affecting both the stock market and the insurance market, see Liang et al [19,20].…”

Section: Introductionmentioning

confidence: 99%

Optimal Reinsurance and Investment under Common Shock Dependence Between Financial and Actuarial Markets

Ceci¹,

Colaneri²,

Cretarola³

2021

Preprint

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We study optimal proportional reinsurance and investment strategies for an insurance company which experiences both ordinary and catastrophic claims and wishes to maximize the expected exponential utility of its terminal wealth. We propose a model where the insurance framework is affected by environmental factors, and aggregate claims and stock prices are subject to common shocks, i.e. drastic events such as earthquakes, extreme weather conditions, or even pandemics, that have an immediate impact on the financial market and simultaneously induce insurance claims. Using the classical stochastic control approach based on the Hamilton-Jacobi-Bellman equation, we provide a verification result for the value function via classical solutions to two backward partial differential equations and characterize the optimal reinsurance and investment strategies. Finally, we make a comparison analysis to discuss the effect of common shock dependence.

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The Optimal Reinsurance-Investment Problem Considering the Joint Interests of an Insurer and a Reinsurer under Hara Utility

Zhang

Zhao²,

Zhou³

2022

Acta Math Sci

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Optimal reinsurance-investment problem with dependent risks based on Legendre transform

Cited by 11 publications

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

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

Optimal Reinsurance and Investment under Common Shock Dependence Between Financial and Actuarial Markets

The Optimal Reinsurance-Investment Problem Considering the Joint Interests of an Insurer and a Reinsurer under Hara Utility

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