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Martingale method for optimal investment and proportional reinsurance

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Applied Mathematics(A Journal of Chinese Universities) 2021年第1期36卷 16-30页

作者： LIU Shuang-sui GUO Wen-jing TONG Xin-le School of Finance Nanjing University of Finance and EconomicsNanjing 210023China.

Numerous researchers have applied the martingale approach for models driven by L¶evy processes to study optimal investment problems.This paper considers an insurer who wants to maximize the expected utility of terminal wealth by selecting optimal investment and proportional reinsurance strategies.The insurer's risk process is modeled by a L¶evy process and the capital can be invested in a security market described by the standard Black-Scholes model.By the martingale approach,the closed-form solutions to the problems of expected utility maximization are derived.Numerical examples are presented to show the impact of model parameters on the optimal strategies.

关键词： martingale method proportional reinsurance investment exponential utility quadratic utility.

Optimal proportional reinsurance for Controlled Risk Process which is Perturbed by Diffusion

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Acta Mathematicae Applicatae Sinica 2007年第3期23卷 477-488页

作者： Zhi-bin Liang Institute of Mathematics and Computer Science Nanjing Normal University Nanjing 210097 China School of Mathematical Sciences and LPMC Nankai University Tianjin 300071 China

In this paper, we study optimal proportional reinsurance policy of an insurer with a risk process which is perturbed by a diffusion. We derive closed-form expressions for the policy and the value function, which are optimal in the sense of maximizing the expected utility in the jump-diffusion framework. We also obtain explicit expressions for the policy and the value function, which are optimal in the sense of maximizing the expected utility or maximizing the survival probability in the diffusion approximation case. Some numerical examples are presented, which show the impact of model parameters on the policy. We also compare the results under the different criteria and different cases.

关键词： Stochastic control Hamilton-Jacobi-Bellman equation jump-diffusion brownian motion diffusion approximation proportional reinsurance

Optimal Dividend and Dynamic reinsurance Strategies with Capital Injections and proportional Costs

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Acta Mathematicae Applicatae Sinica 2012年第3期28卷 505-524页

作者： Yi-dong WU Jun-yi GUO School of Mathematical Sciences and LPMC Nankai UniversityTianjin 300071China School of Economics Yunnan UniversityKunming 650091China

We consider an optimization problem of an insurance company in the diffusion setting, which controls the dividends payout as well as the capital injections. To maximize the cumulative expected discounted dividends minus the penalized discounted capital injections until the ruin time, there is a possibility of （cheap or non-cheap） proportional reinsurance. We solve the control problems by constructing two categories of suboptimal models, one without capital injections and one with no bankruptcy by capital injection. Then we derive the explicit solutions for the value function and totally characterize the optimal strategies. Particularly, for cheap reinsurance, they axe the same as those in the model of no bankruptcy.

关键词： proportional reinsurance non-cheap optimal dividend capital injections

Optimal Dividend and Capital Injection of the Insurance Company with proportional reinsurance Policy

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Optimal Dividend and Capital Injection of the Insurance Comp...

Optimal reinsurance and Dividend Under Model Uncertainty

第二十九届中国控制会议

作者： LIU Guoxin1,ZHANG Shuaiqi2 1.School of Sciences,Heibei University of Technology,Tianjin,300130,ChinaSchool of Mathematical Science and Computing Technology,Central South University,Changsha,Hunan 410075,P.R.China2.School of Mathematical Science and Computing Technology,Central South University,Changsha,Hunan 410075,P.R.China

This paper deals with the optimal control problem of the insurance company with proportional reinsurance policy.By controlling the dynamic reinsurance rate,dividends payout as well as the equity issuance processes,we maximize the cumulative expected discounted dividends payout minus the equity issuance.Moreover,we prove that it is optimal to issue new equity when the process hits 0.

关键词： Optimal dividend control Capital injection proportional reinsurance HJB equation Mixed singular-regular control Stochastic control

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Journal of Systems Science & Complexity 2023年第3期36卷 1116-1143页

作者： LIU Jingzhen WANG Yike ZHANG Ning China Institute for Actuarial Science Central University of Finance and EconomicsBeijing 100081China School of Finance&Chinese Fintech Research Center Central University of Finance and EconomicsBeijing 100081China

In this paper,the authors analyze the optimal reinsurance and dividend problem with model uncertainty for an insurer.Here the model uncertainty represents possible deviations between the real market and the assumed model.In addition to the incorporation of model uncertainty into the traditional diffusion surplus process,the authors include a penalty function in the objective function.The proposed goal is to find the optimal reinsurance and dividend strategy that maximizes the expected discounted dividend before ruin in the worst case of all possible scenarios,namely,the worst market.Using a dynamic programming approach,the problem is reduced to solving a Hamilton-Jacob-Bellman-Isaac(HJBI)equation with singular control.This problem is more difficult than the traditional robust control or singular control problem.Here,the authors prove that the value function is the unique solution to this HJBI equation with singular control.Moreover,the authors present a verification theorem when a smooth solution can be found,and derive closed-form solution when the function in the objective function is specified.

关键词： Hamilton-Jacobi-Bellman-Isaac equation model uncertainty optimal dividend proportional reinsurance

THE OPTIMAL STRATEGY FOR INSURANCE COMPANY UNDER THE INFLUENCE OF TERMINAL VALUE

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Acta Mathematica Scientia 2011年第3期31卷 1077-1090页

作者：刘伟袁海丽胡亦钧 School of Mathematics and Statistics Wuhan University School of Mathematics and System Sciences Xinjiang University

This paper considers a model of an insurance company which is allowed to invest a risky asset and to purchase proportional reinsurance. The objective is to find the policy which maximizes the expected total discounted dividend pay-out until the time of bankruptcy and the terminal value of the company under liquidity constraint. We find the solution of this problem via solving the problem with zero terminal value. We also analyze the influence of terminal value on the optimal policy.

关键词： proportional reinsurance terminal value optimal policy HJB equation

Optimal reinsurance and Investment Strategy with Delay in Heston’s SV Model

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Journal of the Operations Research Society of China 2021年第2期9卷 245-271页

作者： Chun-Xiang A Ai-Lin Gu Yi Shao School of Mathematics and Statistics Zhaoqing UniversityZhaoqing 526061GuangdongChina School of Applied Mathematics Guangdong University of TechnologyGuangzhou 510520China

In this paper,we consider an optimal investment and proportional reinsurance problem with delay,in which the insurer’s surplus process is described by a jump-diffusion model.The insurer can buy proportional reinsurance to transfer part of the insurance claims risk.In addition to reinsurance,she also can invests her surplus in a financial market,which is consisted of a risk-free asset and a risky asset described by Heston’s stochastic volatility(SV)model.Considering the performance-related capital flow,the insurer’s wealth process is modeled by a stochastic differential delay equation.The insurer’s target is to find the optimal investment and proportional reinsurance strategy to maximize the expected exponential utility of combined terminal wealth.We explicitly derive the optimal strategy and the value function.Finally,we provide some numerical examples to illustrate our results.

关键词： proportional reinsurance Stochastic differential delay equation(SDDE) Heston’s stochastic volatility(SV)model Hamilton–Jacobi–Bellman(HJB)equation

Optimal reinsurance and Investment Policies with the CEV Stock Market

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Acta Mathematicae Applicatae Sinica 2016年第3期32卷 647-658页

作者： Qi-cai LI Meng-di GU School of Mathematical Sciences Nanjing Normal University Antai College of Economics and Management Shanghai Jiaotong University

In this paper, under the criterion of maximizing the expected exponential utility of terminal wealth, we study the optimal proportional reinsurance and investment policy for an insurer with the compound Poisson claim process. We model the price process of the risky asset to the constant elasticity of variance （for short, CEV） model, and consider net profit condition and variance reinsurance premium principle in our work. Using stochastic control theory, we derive explicit expressions for the optimal policy and value function. And some numerical examples are given.

关键词： proportional reinsurance CEV model stochastic control Hamilton-Jacobi-Bellman equation exponential utility

The Optimal Policy for Insurance Company Under Consideration of Internal Competition and the Time Value of Ruin

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Acta Mathematicae Applicatae Sinica 2014年第3期30卷 807-818页

作者： Wei LIU Yi-jun HU School of Mathematics and System Sciences Xinjiang University School of Mathematics and Statistics Wuhan University

This paper considers the dividend optimization problem for an insurance company under the consideration of internal competition between different units inside the company. The objective is to find a reinsurance policy and a dividend payment scheme so as to maximize the expected discounted value of the dividend payment, and the expected present value of an amount which the insurer earns until the time of ruin. By solving the corresponding constrained Hamilton-Jacobi-Bellman （HJB） equation, we obtain the value function and the optimal reinsurance policy and dividend payment.

关键词： dividend payment proportional reinsurance internal competition Hamilton-Jacobi-Bellman （HJB）equation