相依风险模型下具有延迟和违约风险的鲁棒最优投资和再保险策略（英文）

本文研究了在风险相依模型下具有延迟和违约风险的鲁棒最优投资再保险策略.假设模糊厌恶型保险人的财富过程有两类相依的保险业务并且余额可以投资于无风险资产、可违约债券和价格过程遵循Heston模型的风险资产.利用动态规划原则,我们分别建立了违约后和违约前的鲁棒HJB方程.另外,通过最大化终端财富的期望指数效用,我们得到了最优投资和再保险策略以及相应的值函数.最后,通过一些数值例子说明了某些模型参数对鲁棒最优策略的影响.     

风险相依下再保险双方的联合最优再保险问题

结合保险人和再保险人的共同利益,研究了具有两类相依险种风险模型下的最优再保险问题.假定再保险公司采用方差保费原理收取保费,利用复合Poisson模型和扩散逼近模型两种方式去刻画保险公司和再保险公司的资本盈余过程,在期望效用最大准则下,证明了最优再保险策略的存在性和唯一性,通过求解Hamilton-Jacobi-Bellman（HJB）方程,得到了两种模型下相应的最优再保险策略及值函数的明晰解答,并给出了数值算例及分析.     

风险相依下再保险双方的联合最优再保险问题

结合保险人和再保险人的共同利益,研究了具有两类相依险种风险模型下的最优再保险问题.假定再保险公司采用方差保费原理收取保费,利用复合Poisson模型和扩散逼近模型两种方式去刻画保险公司和再保险公司的资本盈余过程,在期望效用最大准则下,证明了最优再保险策略的存在性和唯一性,通过求解Hamilton-Jacobi-Bellman(HJB)方程,得到了两种模型下相应的最优再保险策略及值函数的明晰解答,并给出了数值算例及分析.     

汇率风险和方差保费准则下最优投资和再保险策略

假定保险公司和再保险公司都采取方差保费准则收取保费,保险公司不但可以投资本国无风险资产和风险资产,还可以投资国外的风险资产.首先我们用一几何布朗运动来刻画汇率风险,同时为了控制保险风险,假定保险公司将承担的保险业务分保给再保险公司.接着利用随机动态规划原理研究了两种情形下的最优投资和再保险问题,一种是索赔服从扩散近似模型;另一种是经典风险模型,分别得到了这两种情形下的最优投资和再保险策略,并发现汇率风险对保险公司的投资策略有很大的影响,但对再保险策略没有影响.最后对相关参数进行了敏感性分析.     

指数保费准则下的最优投资和比例再保险

该文研究了保险公司的最优投资和比例再保险问题,其中假定保险公司的盈余过程为一个带扩散扰动的经典风险过程.假定再保险的保费按照指数保费原理来计算,这使得所研究的随机控制问题成为非线性的.该文同时考虑了最大化终端财富指数效用和最大化调节系数两类问题,并给出了最优值函数和相应的最优策略的解析表达.此外,该文还分析了再保险公司的风险厌恶和保险公司的不确定性参数对最优策略的影响.     

基于损失相依保费原则下的最优再保险-投资策略

本文研究了基于损失相依保费原则下的最优再保险投资问题。该保费原则是基于过去的损失和对未来损失的估计来动态地更新保费，是传统的期望值保费原则的一个拓展。我们假设保险公司的盈余过程遵循C-L(Cramér-Lundberg)模型的扩散近似，保险公司通过购买比例再保险或获得新业务来分散风险或增加收益。假设金融市场由一个无风险资产和一个风险资产组成，其中风险资产的价格过程由仿射平方根随机模型描述。我们以最大化保险公司的终端时刻财富的期望效用为目标，利用动态规划，随机控制等方法得到CARA效用函数下的值函数的解析解，并得到最优再保险和投资策略的显性表达式。最后通过数值算例，分析了部分模型参数对最优再保险投资策略的影响。     

CEV模型下时滞最优投资与再保险问题

在常方差弹性（constant elasticity of variance,CEV）模型下考虑了时滞最优投资与比例再保险问题.假设保险公司通过购买比例再保险对保险索赔风险进行管理,并将其财富投资于一个无风险资产和一个风险资产组成的金融市场,其中风险资产的价格过程服从常方差弹性模型.考虑与历史业绩相关的现金流量,保险公司的财富过程由一个时滞随机微分方程刻画,在负指数效用最大化的目标下求解了时滞最优投资与再保险控制问题,分别在投资与再保险和纯投资两种情形下得到最优策略和值函数的解析表达式.最后通过数值算例进一步说明主要参数对最优策略和值函数的影响.     

CEV模型下时滞最优投资与再保险问题

在常方差弹性(constant elasticity of variance,CEV)模型下考虑了时滞最优投资与比例再保险问题.假设保险公司通过购买比例再保险对保险索赔风险进行管理,并将其财富投资于一个无风险资产和一个风险资产组成的金融市场,其中风险资产的价格过程服从常方差弹性模型.考虑与历史业绩相关的现金流量,保险公司的财富过程由一个时滞随机微分方程刻画,在负指数效用最大化的目标下求解了时滞最优投资与再保险控制问题,分别在投资与再保险和纯投资两种情形下得到最优策略和值函数的解析表达式.最后通过数值算例进一步说明主要参数对最优策略和值函数的影响.     

违约市场中的n个保险公司的最优投资和再保险问题（英文）

本文研究了n个保险公司之间的非零和随机微分投资再保险博弈问题.每个保险公司可以购买比例再保险,并将财富投资于一个由无风险资产,可违约债券和n个风险资产组成的金融市场.特别地,风险资产的价格过程服从CEV模型,可违约债券可在违约时收回一定比例的价值.每个保险公司的目标是相对于竞争对手,最大化终端财富的期望指数效用.利用随机最优控制理论,我们分别推导了均衡策略和均衡值函数的显式表达式.数值例子分析了模型参数对均衡策略的影响.此外,我们还分析了保险公司数量对均衡投资策略的影响.我们发现,随着保险公司数量的增加,每个保险公司将在风险资产和可违约债券上投入更多的资金.     

王氏保费准则下隐含再保险公司违约风险的最优再保险设计

本文考虑到再保险公司违约风险对保险人再保险的影响,利用VaR风险度量研究最优再保险策略.在再保险合同中,再保险公司向保险人收取一定的保费,承诺赔偿再保险人面临的部分损失.但,当再保险公司承诺的限额超过其偿付能力就可能发生违约风险.因此,为了避免再保险公司违约风险,使保险公司的总风险最小,本文根据王氏保费准则,运用VaR风险度量的最优化标准,得到分层再保险是最优的,并给出相应的数值算例.     

Optimal reinsurance with regulatory initial capital and default risk

In a reinsurance contract, a reinsurer promises to pay the part of the loss faced by an insurer in exchange for receiving a reinsurance premium from the insurer. However, the reinsurer may fail to pay the promised amount when the promised amount exceeds the reinsurer’s solvency. As a seller of a reinsurance contract, the initial capital or reserve of a reinsurer should meet some regulatory requirements. We assume that the initial capital or reserve of a reinsurer is regulated by the value-at-risk (VaR) of its promised indemnity. When the promised indemnity exceeds the total of the reinsurer’s initial capital and the reinsurance premium, the reinsurer may fail to pay the promised amount or default may occur. In the presence of the regulatory initial capital and the counterparty default risk, we investigate optimal reinsurance designs from an insurer’s point of view and derive optimal reinsurance strategies that maximize the expected utility of an insurer’s terminal wealth or minimize the VaR of an insurer’s total retained risk. It turns out that optimal reinsurance strategies in the presence of the regulatory initial capital and the counterparty default risk are different both from optimal reinsurance strategies in the absence of the counterparty default risk and from optimal reinsurance strategies in the presence of the counterparty default risk but without the regulatory initial capital.     

OPTIMAL PROPORTIONAL REINSURANCE AND INVESTMENT FOR A CONSTANT ELASTICITY OF VARIANCE MODEL UNDER VARIANCE PRINCIPLE

This article studies the optimal proportional reinsurance and investment problem under a constant elasticity of variance(CEV) model.Assume that the insurer’s surplus process follows a jump-diffusion process,the insurer can purchase proportional reinsurance from the reinsurer via the variance principle and invest in a risk-free asset and a risky asset whose price is modeled by a CEV model.The diffusion term can explain the uncertainty associated with the surplus of the insurer or the additional small claims.The objective of the insurer is to maximize the expected exponential utility of terminal wealth.This optimization problem is studied in two cases depending on the diffusion term’s explanation.In all cases,by using techniques of stochastic control theory,closed-form expressions for the value functions and optimal strategies are obtained.     

A Bowley solution with limited ceded risk for a monopolistic reinsurer

Borch (1969) advocated that the study of optimal reinsurance design should take into consideration the conflicting interests of both an insurer and a reinsurer. Motivated by this and exploiting a Bowley solution (or Stackelberg equilibrium game), this paper proposes a two-step model that tackles an optimal risk transfer problem between the insurer and the reinsurer. From the insurer’s perspective, the first step of the model provisionally derives an optimal reinsurance policy for a given reinsurance premium while reflecting the reinsurer’s risk appetite. The reinsurer’s risk appetite is controlled by imposing upper limits on the first two moments of the coverage. Through a comparative analysis, the effect of the insurer’s initial wealth on the demand for reinsurance is then examined, when the insurer’s risk aversion and prudence are taken into account. Based on the insurer’s provisional strategy, the second step of the model determines the monopoly premium that maximizes the reinsurer’s expected profit while still satisfying the insurer’s incentive condition. Numerical examples are provided to illustrate our Bowley solution.     

Optimal reinsurance policy: The adjustment coefficient and the expected utility criteria

This paper is concerned with the optimal form of reinsurance from the ceding company point of view, when the cedent seeks to maximize the adjustment coefficient of the retained risk. We deal with the problem by exploring the relationship between maximizing the adjustment coefficient and maximizing the expected utility of wealth for the exponential utility function, both with respect to the retained risk of the insurer.Assuming that the premium calculation principle is a convex functional and that some other quite general conditions are fulfilled, we prove the existence and uniqueness of solutions and provide a necessary optimal condition. These results are used to find the optimal reinsurance policy when the reinsurance premium calculation principle is the expected value principle or the reinsurance loading is an increasing function of the variance. In the expected value case the optimal form of reinsurance is a stop-loss contract. In the other cases, it is described by a nonlinear function.     

Markov调节中基于时滞和相依风险模型的最优再保险与投资

本文研究保险公司在Markov调节下基于时滞及相依风险模型的最优再保险与最优投资问题,其中市场被划分为有限个状态,一些重要的参数随着市场状态的转换而变化.假设保险公司的盈余过程由复合Poisson过程描述,而风险资产的价格过程由几何跳扩散模型刻画,并且假设这两个跳过程是相依的.以最大化终端财富值的均值-方差效用为目标,在博弈论框架下,利用随机控制理论和相应的广义Hamilton-Jacobi-Bellman(HJB)方程,本文得到最优策略和值函数的显式表达,并证明解的存在性和唯一性.最后,通过一些数值实例,验证所得结论的正确性,并探讨一些重要参数对最优策略的影响.     

Exponential utility maximization for an insurer with time-inconsistent preferences

This paper studies the optimal consumption–investment–reinsurance problem for an insurer with a general discount function and exponential utility function in a non-Markovian model. The appreciation rate and volatility of the stock, the premium rate and volatility of the risk process of the insurer are assumed to be adapted stochastic processes, while the interest rate is assumed to be deterministic. The object is to maximize the utility of intertemporal consumption and terminal wealth. By the method of multi-person differential game, we show that the time-consistent equilibrium strategy and the corresponding equilibrium value function can be characterized by the unique solutions of a BSDE and an integral equation. Under appropriate conditions, we show that this integral equation admits a unique solution. Furthermore, we compare the time-consistent equilibrium strategies with the optimal strategy for exponential discount function, and with the strategies for naive insurers in two special cases.     

带跳一扩散风险过程保险人的最优控制（英文）

In this paper, the optimal XL-reinsurance of an insurer with jump-diffusion risk process is studied. With the assumptions that the risk process is a compound Possion process perturbed by a standard Brownian motion and the reinsurance premium is calculated according to the variance principle, the implicit expression of the priority and corresponding value function when the utility function is exponential are obtained. At last, the value function is argued, the properties of the priority about parameters are discussed and numerical results of the priority for various claim-size distributions are shown.     

Optimal reinsurance and investment policies with the CEV stock market

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.