状态相依效用下的超额损失再保险——投资策略

假设保险公司的盈余过程和金融市场的资产价格过程均由可观测的连续时间马尔科夫链所调节, 以最大化终端财富的状态相依的期望指数效用为目标, 研究了保险公司的超额损失再保险-投资问题. 运用动态规划方法, 得到最优再保险-投资策略的解析解以及最优值函数的半解析式. 最后, 通过数值例子, 分析了模型各参数对最优值函数和最优策略的影响.     

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。     

基于MAP风险模型的最优分红问题研究

在保险公司财务核算和分红均发生在随机时间点的假设条件下,讨论保险公司的最优分红问题.假设保险公司的盈余过程是经过MAP(马氏到达过程)的相过程调制的复合泊松过程,保险公司对盈余过程的观测和分红都发生在MAP的跳点上,以最大化期望折现分红总量为目标,证明了最优分红策略为band策略,并分析了经济状态和分红机会对值函数和分红策略的影响.     

在不允许卖空条件下的最优比例再保险投资

假设保险公司的盈余过程服从一个带扰动项的布朗运动,保险公司可以投资一个无风险资产和n个风险资产,还可以购买比例再保险,并且风险市场是不允许卖空的.本文在均值一方差优化准则下研究保险公司的最优投资一再保策略选择问题,利用LQ随机控制方法求解模型,得到了保险公司的最优组合投资策略的解析和保险公司投资的有效投资边界的解析表达...     

具有随机利率、随机变差的最优投资和联合比例-超额损失再保险

对跳-扩散风险模型,研究了最优投资和再保险问题.保险公司可以购买再保险减少理赔,保险公司还可以把盈余投资在一个无风险资产和一个风险资产上.假设再保险的方式为联合比例-超额损失再保险.还假设无风险资产和风险资产的利率是随机的,风险资产的方差也是随机的.通过解决相应的Hamilton-Jacobi-Bellman(HJB)方程,获得了最优值函数和最优投资、再保险策略的显示解.特别的,通过一个例子具体的解释了得到的结论.     

正相依风险下离散模型的最优分红

将保险公司各期净损失相互独立的假定改进为依随机序正相依.在相依风险下,利用动态规划原理和状态空间约简,刻画了最优分红策略,证明了区域策略最优,同时讨论了值函数的性质,并给出了数值算法.其中,对涉及独立假定的结论,给出了相依条件下的相应结果,对未涉及独立假定的部分结论也做了改进.研究发现,与独立情形不同,在依随机序正相依风险下,保险公司不必以概率1破产.     

带交易费用的最优投资和比例再保险

研究了保险公司的最优投资和再保险问题.保险公司的盈余通过跳-扩散风险模型来模拟,可以把盈余的一部分投资到金融市场,金融市场由一个无风险资产和n个风险资产组成,并且保险公司还可以购买比例再保险;在买卖风险资产时,考虑了交易费用.通过随机控制的理论,获得了最优策略和值函数的显示解.     

具有共同冲击相依性的跳扩散金融市场中有着延迟和违约风险的鲁棒最优再保险和投资策略

在本文中,我们考虑跳扩散模型下具有延迟和违约风险的鲁棒最优再保险和投资问题,保险人可以投资无风险资产,可违约的债券和两个风险资产,其中两个风险资产遵循跳跃扩散模型且受到同种因素带来共同影响而相互关联.假设允许保险人购买比例再保险,特别地再保险保费利用均值方差保费原则来计算.在考虑与绩效相关的资本流入/流出下,保险公司的...     

再保险-投资的M-V及M-VaR最优策略

考虑保险公司再保险-投资问题在均值-方差(M-V)模型和均值-在险价值(M-VaR)模型下的最优常数再调整策略.在保险公司盈余过程服从扩散过程的假设及多风险资产的Black-Scholes市场条件下,分别得到均值-方差模型和均值-在险价值模型下保险公司再保险-投资问题的最优常数再调整策略及共有效前沿,并就两种模型下的结...     

Vasicek利率与Heston模型下的最优投资和再保险

本文主要研究Vasicek随机利率模型下保险公司的最优投资与再保险问题.假设保险公司的盈余过程由带漂移的布朗运动来描述,保险公司通过购买比例再保险来转移索赔风险;同时,将财富投资于由一种无风险资产与一种风险资产组成的金融市场,其中,利率期限结构服从Vasicek利率模型,且风险资产价格过程满足Heston随机波动率模型.利用动态规划原理及变量替换的方法,得到了指数效用下最优投资与再保险策略的显示表达式,并给出数值例子分析了主要模型参数对最优策略的影响.     

Optimal time-consistent investment and reinsurance policies for mean-variance insurers

This paper investigates the optimal time-consistent policies of an investment-reinsurance problem and an investment-only problem under the mean-variance criterion for an insurer whose surplus process is approximated by a Brownian motion with drift. The financial market considered by the insurer consists of one risk-free asset and multiple risky assets whose price processes follow geometric Brownian motions. A general verification theorem is developed, and explicit closed-form expressions of the optimal polices and the optimal value functions are derived for the two problems. Economic implications and numerical sensitivity analysis are presented for our results. Our main findings are: (i) the optimal time-consistent policies of both problems are independent of their corresponding wealth processes; (ii) the two problems have the same optimal investment policies; (iii) the parameters of the risky assets (the insurance market) have no impact on the optimal reinsurance (investment) policy; (iv) the premium return rate of the insurer does not affect the optimal policies but affects the optimal value functions; (v) reinsurance can increase the mean-variance utility.     

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.     

Pricing general insurance in a reactive and competitive market

A simple parameterisation is introduced which represents the insurance market’s response to an insurer adopting a pricing strategy determined via optimal control theory. Claims are modelled using a lognormally distributed mean claim size rate, and the market average premium is determined via the expected value principle. If the insurer maximises its expected wealth then the resulting Bellman equation has a moving boundary in state space that determines when it is optimal to stop selling insurance. This stochastic optimisation problem is simplified by the introduction of a stopping time that prevents an insurer leaving and then re-entering the insurance market. Three finite difference schemes are used to verify the existence of a solution to the resulting Bellman equation when there is market reaction. All of the schemes use a front-fixing transformation. If the market reacts, then it is found that the optimal strategy is altered, in that premiums are raised if the strategy is of loss-leading type and lowered if it is optimal for the insurer to set a relatively high premium and sell little insurance.     

Robust optimal investment and reinsurance problem for a general insurance company under Heston model

In this paper, we study a robust optimal investment and reinsurance problem for a general insurance company which contains an insurer and a reinsurer. Assume that the claim process described by a Brownian motion with drift, the insurer can purchase proportional reinsurance from the reinsurer. Both the insurer and the reinsurer can invest in a financial market consisting of one risk-free asset and one risky asset whose price process is described by the Heston model. Besides, the general insurance company’s manager will search for a robust optimal investment and reinsurance strategy, since the general insurance company faces model uncertainty and its manager is ambiguity-averse in our assumption. The optimal decision is to maximize the minimal expected exponential utility of the weighted sum of the insurer’s and the reinsurer’s surplus processes. By using techniques of stochastic control theory, we give sufficient conditions under which the closed-form expressions for the robust optimal investment and reinsurance strategies and the corresponding value function are obtained.     

Pricing general insurance in a reactive and competitive market

A simple parameterisation is introduced which represents the insurance market’s response to an insurer adopting a pricing strategy determined via optimal control theory. Claims are modelled using a lognormally distributed mean claim size rate, and the market average premium is determined via the expected value principle. If the insurer maximises its expected wealth then the resulting Bellman equation has a moving boundary in state space that determines when it is optimal to stop selling insurance. This stochastic optimisation problem is simplified by the introduction of a stopping time that prevents an insurer leaving and then re-entering the insurance market. Three finite difference schemes are used to verify the existence of a solution to the resulting Bellman equation when there is market reaction. All of the schemes use a front-fixing transformation. If the market reacts, then it is found that the optimal strategy is altered, in that premiums are raised if the strategy is of loss-leading type and lowered if it is optimal for the insurer to set a relatively high premium and sell little insurance.     

Time-consistent mean–variance proportional reinsurance and investment problem in a defaultable market

In this paper, we consider an optimal time-consistent reinsurance-investment problem incorporating a defaultable security for a mean–variance insurer under a constant elasticity of variance (CEV) model. In our model, the insurer’s surplus process is described by a jump-diffusion risk model, the insurer can purchase proportional reinsurance and invest in a financial market consisting of a risk-free asset, a defaultable bond and a risky asset whose price process is assumed to follow a CEV model. Using a game theoretic approach, we establish the extended Hamilton–Jacobi–Bellman system for the post-default case and the pre-default case, respectively. Furthermore, we obtain the closed-from expressions for the time-consistent reinsurance-investment strategy and the corresponding value function in both cases. Finally, we provide numerical examples to illustrate the impacts of model parameters on the optimal time-consistent strategy.     

Optimal investment and risk control policies for an insurer in an incomplete market

In this paper, we apply the martingale approach to investigate the optimal investment and risk control problem for an insurer in an incomplete market. The claim risk of per policy is characterized by a compound Poisson process with drift, and the insurer can be invested in multiple risky assets whose price processes are described by the geometric Brownian motions model. By ‘complete’ the incomplete market, closed-form solutions to the problems of mean–variance criterion and expected exponential utility maximization are obtained. Moreover, numerical simulations are presented to illustrate the results with the basic parameters.     

Time-consistent reinsurance and investment strategies for mean–variance insurer under partial information

In this paper, based on equilibrium control law proposed by Björk and Murgoci (2010), we study an optimal investment and reinsurance problem under partial information for insurer with mean–variance utility, where insurer’s risk aversion varies over time. Instead of treating this time-inconsistent problem as pre-committed, we aim to find time-consistent equilibrium strategy within a game theoretic framework. In particular, proportional reinsurance, acquiring new business, investing in financial market are available in the market. The surplus process of insurer is depicted by classical Lundberg model, and the financial market consists of one risk free asset and one risky asset with unobservable Markov-modulated regime switching drift process. By using reduction technique and solving a generalized extended HJB equation, we derive closed-form time-consistent investment–reinsurance strategy and corresponding value function. Moreover, we compare results under partial information with optimal investment–reinsurance strategy when Markov chain is observable. Finally, some numerical illustrations and sensitivity analysis are provided.     

Optimal investment–reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets

In this paper, we investigate the optimal time-consistent investment–reinsurance strategies for an insurer with state dependent risk aversion and Value-at-Risk (VaR) constraints. The insurer can purchase proportional reinsurance to reduce its insurance risks and invest its wealth in a financial market consisting of one risk-free asset and one risky asset, whose price process follows a geometric Brownian motion. The surplus process of the insurer is approximated by a Brownian motion with drift. The two Brownian motions in the insurer’s surplus process and the risky asset’s price process are correlated, which describe the correlation or dependence between the insurance market and the financial market. We introduce the VaR control levels for the insurer to control its loss in investment–reinsurance strategies, which also represent the requirement of regulators on the insurer’s investment behavior. Under the mean–variance criterion, we formulate the optimal investment–reinsurance problem within a game theoretic framework. By using the technique of stochastic control theory and solving the corresponding extended Hamilton–Jacobi–Bellman (HJB) system of equations, we derive the closed-form expressions of the optimal investment–reinsurance strategies. In addition, we illustrate the optimal investment–reinsurance strategies by numerical examples and discuss the impact of the risk aversion, the correlation between the insurance market and the financial market, and the VaR control levels on the optimal strategies.