The existence of weak solutions for a generalized Camassa-Holm equation
Shaoyong Lai Qichang Xie Yunxi Guo YongHong Wu
Communications on Pure & Applied Analysis 2011, 10(1): 45-57 doi: 10.3934/cpaa.2011.10.45
A Camassa-Holm type equation containing nonlinear dissipative effect is investigated. A sufficient condition which guarantees the existence of weak solutions of the equation in lower order Sobolev space $H^s$ with $1 \leq s \leq \frac{3}{2}$ is established by using the techniques of the pseudoparabolic regularization and some prior estimates derived from the equation itself.
keywords: high order nonlinear terms pseudoparabolic regularization technique. weak solution Generalized Camassa-Holm equation
Asset liability management for an ordinary insurance system with proportional reinsurance in a CIR stochastic interest rate and Heston stochastic volatility framework
Yan Zhang Yonghong Wu Benchawan Wiwatanapataphee Francisca Angkola
Journal of Industrial & Management Optimization 2017, 13(5): 1-31 doi: 10.3934/jimo.2018141

This paper investigates the asset liability management problem for an ordinary insurance system incorporating the standard concept of proportional reinsurance coverage in a stochastic interest rate and stochastic volatility framework. The goal of the insurer is to maximize the expectation of the constant relative risk aversion (CRRA) of the terminal value of the wealth, while the goal of the reinsurer is to maximize the expected exponential utility (CARA) of the terminal wealth held by the reinsurer. We assume that the financial market consists of risk-free assets and risky assets, and both the insurer and the reinsurer invest on one risk-free asset and one risky asset. By using the stochastic optimal control method, analytical expressions are derived for the optimal reinsurance control strategy and the optimal investment strategies for both the insurer and the reinsurer in terms of the solutions to the underlying Hamilton-Jacobi-Bellman equations and stochastic differential equations for the wealths. Subsequently, a semi-analytical method has been developed to solve the Hamilton-Jacobi-Bellman equation. Finally, we present numerical examples to illustrate the theoretical results obtained in this paper, followed by sensitivity tests to investigate the impact of reinsurance, risk aversion, and the key parameters on the optimal strategies.

keywords: Asset liability management CIR stochastic interest rate model Heston stochastic volatility model insurance system with reinsurance Hamilton-Jacobi-Bellman equation stochastic optimal control

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