Optimization Problems of Excess-of-Loss Reinsurance and Investment under the CEV Model

Abstract

We consider that the insurer purchases excess-of-loss reinsurance and invests its wealth in the constant elasticity of variance (CEV) stock market. We model risk process by Brownian motion with drift and study the optimization problem of maximizing the exponential utility of terminal wealth under the controls of excess-of-loss reinsurance and investment. Using stochastic control theory and power transformation technique, we obtain explicit expressions for the optimal polices and value function. We also show that the optimal excess-of-loss reinsurance is always better than optimal proportional reinsurance. Some numerical examples are given.

Applications of Stochastic Optimal Control to Economics and Finance

“…The former was intensively studied in Irgens and Paulsen (2004) 2019) and references therein. The latter was investigated in these articles: in Zhang et al (2007) and Meng and Zhang (2010), the authors proved the optimality of the excess-of-loss policy under the criterion of minimizing the ruin probability, with the surplus process described by a Brownian motion with drift; in Zhao et al (2013) the Cramér-Lundberg model is used for the surplus process, with the possibility of investing in a financial market represented by the Heston model; in Sheng et al (2014) and Li and Gu (2013) the risky asset is described by a Constant Elasticity of Variance (CEV) model, while the surplus is modelled by the Cramér-Lundberg model and its diffusion approximation, respectively; finally, in Li et al (2018) the authors studied a robust optimal strategy under the diffusion approximation of the surplus process.…”

Optimal Excess-of-Loss Reinsurance for Stochastic Factor Risk Models

“…The former was intensively studied in Irgens and Paulsen (2004); Liu and Ma (2009); Liang et al (2011); Liang and Bayraktar (2014); Zhu et al (2015); Brachetta and Ceci (2019) and references therein. The latter was investigated in these articles: in Zhang et al (2007) and Meng and Zhang (2010), the authors proved the optimality of the excess-of-loss policy under the criterion of minimizing the ruin probability, with the surplus process described by a Brownian motion with drift; in Zhao et al (2013) the Cramér-Lundberg model is used for the surplus process, with the possibility of investing in a financial market represented by the Heston model; in Sheng et al (2014) and Li and Gu (2013) the risky asset is described by a Constant Elasticity of Variance (CEV) model, while the surplus is modelled by the Cramér-Lundberg model and its diffusion approximation, respectively; finally, in Li et al (2018) the authors studied a robust optimal strategy under the diffusion approximation of the surplus process.…”