# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2019011

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

 1 School of Science, Nanjing University of Science and Technology, Nanjing 210094, China 2 College of Science, Army Engineering University of PLA, Nanjing 211101, China

* Corresponding author

Received  April 2018 Revised  October 2018 Published  March 2019

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

This paper investigates an optimal reinsurance-investment problem in relation to thinning dependent risks. The insurer's wealth process is described by a risk model with two dependent classes of insurance business. The insurer is allowed to purchase reinsurance and invest in one risk-free asset and one risky asset whose price follows CEV model. Our aim is to maximize the expected exponential utility of terminal wealth. Applying Legendre transform-dual technique along with stochastic control theory, we obtain the closed-form expression of optimal strategy. In addition, our wealth process will reduce to the classical Cramér-Lundberg (C-L) model when $p = 0$, in this case, we achieve the explicit expression of the optimal strategy for Hyperbolic Absolute Risk Aversion (HARA) utility by using Legendre transform. Finally, some numerical examples are presented to illustrate the impact of our model parameters (e.g., interest and volatility) on the optimal reinsurance-investment strategy.

Citation: Yan Zhang, Peibiao Zhao. Optimal reinsurance-investment problem with dependent risks based on Legendre transform. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019011
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##### References:
Effect of $t$ on the optimal reinsurance strategies
Effect of $v$ on the optimal reinsurance strategies
Effect of $p$ on the optimal reinsurance strategies
Effect of $\alpha _1$ on the optimal reinsurance strategies
Effect of $\alpha _2$ on the optimal reinsurance strategies
Effect of $s$ on the optimal investment strategy
Effect of $\mu - r$ on the optimal investment strategy
Effect of $\sigma$ on the optimal investment strategy
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