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.     

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.     

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.     

基于Volterra Heston模型的最优投资和再保险问题

作为金融市场体系的重要组成部分,选择最优的投资和再保险策略对保险公司来说十分重要.本文研究了保险公司在均值-方差准则下的最优投资和再保险问题,假设保险公司通过购买比例再保险来分散自身风险,其盈余过程由近似经典Cramer-Lundberg模型的扩散过程刻画,此外,保险公司通过投资于无风险资产和风险资产来增加收入,其中风险资产价格服从Volterra Heston模型.由于Volterra Heston模型的非马尔可夫性和非半鞅性,经典的随机最优控制框架不再适用,本文通过构造一个辅助随机过程,得到了依赖于Riccati-Volterra方程解的最优投资和再保险策略及有效前沿,并对最优策略、有效前沿和波动率粗糙度、再保险因素之间的关系进行了数值分析,发现股票价格的波动率越粗糙,保险公司对股票市场和再保险的需求越大.     

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.     

考虑交易费用和债务的再保险和投资问题

该文考虑了保险公司的再保险和投资在多种风险资产中的策略问题. 假设保险公司本身有着一定的债务, 债务的多少服从线性扩散方程. 保险公司可以通过再保险和将再保险之后的剩余资产投资在m种风险资产和一种无风险资产中降低其风险. 资产中风险资产的价格波动服从几何布朗运动, 其债务多少的演化也是依据布朗运动而上下波动. 该文考虑了风险资产与债务之间的相互关系, 考虑了在进行风险投资时的交易费用, 并且利用HJB方程求得保险公司的最大最终资产的预期指数效用, 给出了相应的最优价值函数和最优策略的数值解.     

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.     

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.     

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,保险公司的财富过程通过随机微分延迟方程建模.保险公司的目标是最大程度地发挥终端财富和平均绩效财富组合的预期指数效用,以分别研究违约前和违约后的情况.此外,推导了最优策略的闭式表达式和相应的价值函数.最后通过数值算例和敏感性分析,表明了各种参数对最优策略的影响.另外对于模糊厌恶投资者,忽视模型模糊性风险会带来显著的效用损失.     

Poisson-Geometric模型下时间一致的最优再保险-投资策略选择

本文研究Poisson-Geometric模型下,时间一致的再保险-投资策略选择问题.在风险模型中,理赔发生次数用Poisson-Geometric过程描述,保险公司在进行再保险时,按照方差值原理计算再保险的保费.保险人在金融市场上投资时,风险资产满足带跳的随机微分方程.保险人的目标是,选择一个时间一致的再保险-投资策略,最大化终止时刻财富的均值同时最小化其方差.通过使用随机控制理论,求得时间一致的再保险-投资策略以及值函数的显式解.最后分析结果的经济意义,并通过数值计算,解释了模型参数对最优策略的影响.     

Robust investment–reinsurance optimization with multiscale stochastic volatility

This paper investigates the investment and reinsurance problem in the presence of stochastic volatility for an ambiguity-averse insurer (AAI) with a general concave utility function. The AAI concerns about model uncertainty and seeks for an optimal robust decision. We consider a Brownian motion with drift for the surplus of the AAI who invests in a risky asset following a multiscale stochastic volatility (SV) model. We formulate the robust optimal investment and reinsurance problem for a general class of utility functions under a general SV model. Applying perturbation techniques to the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation associated with our problem, we derive an investment–reinsurance strategy that well approximates the optimal strategy of the robust optimization problem under a multiscale SV model. We also provide a practical strategy that requires no tracking of volatility factors. Numerical study is conducted to demonstrate the practical use of theoretical results and to draw economic interpretations from the robust decision rules.     

Optimal excess-of-loss reinsurance and investment problem for an insurer with jump–diffusion risk process under the Heston model

In this paper, we study the optimal excess-of-loss reinsurance and investment problem for an insurer with jump–diffusion risk model. The insurer is allowed to purchase reinsurance and invest in one risk-free asset and one risky asset whose price process satisfies the Heston model. The objective of the insurer is to maximize the expected exponential utility of terminal wealth. By applying stochastic optimal control approach, we obtain the optimal strategy and value function explicitly. In addition, a verification theorem is provided and the properties of the optimal strategy are discussed. Finally, we present a numerical example to illustrate the effects of model parameters on the optimal investment–reinsurance strategy and the optimal value function.     

Time-consistent investment-proportional reinsurance strategy with random coefficients for mean–variance insurers

In this study, we consider an insurer who manages her underlying risk by purchasing proportional reinsurance and investing in a financial market consisting of a risk-free bond and a risky asset. The objective of the insurer is to identify an investment–reinsurance strategy that minimizes the mean–variance cost function. We obtain a time-consistent open-loop equilibrium strategy and the corresponding efficient frontier in explicit form using two systems of backward stochastic differential equations. Furthermore, we apply our results to Vasiček’s stochastic interest rate model and Heston’s stochastic volatility model. In both cases, we obtain a closed-form solution.     

Optimal Mean-Variance Problem with Constrained Controls in a Jump-Diffusion Financial Market for an Insurer

In this paper, we study the optimal investment and optimal reinsurance problem for an insurer under the criterion of mean-variance. The insurer’s risk process is modeled by a compound Poisson process and the insurer can invest in a risk-free asset and a risky asset whose price follows a jump-diffusion process. In addition, the insurer can purchase new business (such as reinsurance). The controls (investment and reinsurance strategies) are constrained to take nonnegative values due to nonnegative new business and no-shorting constraint of the risky asset. We use the stochastic linear-quadratic (LQ) control theory to derive the optimal value and the optimal strategy. The corresponding Hamilton–Jacobi–Bellman (HJB) equation no longer has a classical solution. With the framework of viscosity solution, we give a new verification theorem, and then the efficient strategy (optimal investment strategy and optimal reinsurance strategy) and the efficient frontier are derived explicitly.     

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.     

基于新巴塞尔协议监管下保险人的均值-方差最优投资-再保险问题

本文研究了均值-方差优化准则下,保险人的最优投资和最优再保险问题.我们用一个复合泊松过程模型来拟合保险人的风险过程,保险人可以投资无风险资产和价格服从跳跃-扩散过程的风险资产.此外保险人还可以购买新的业务(如再保险).本文的限制条件为投资和再保险策略均非负,即不允许卖空风险资产,且再保险的比例系数非负.除此之外,本文还引入了新巴塞尔协议对风险资产进行监管,使用随机二次线性(linear-quadratic,LQ)控制理论推导出最优值和最优策略.对应的哈密顿-雅克比-贝尔曼(Hamilton-Jacobi-Bellman,HJB)方程不再有古典解.在粘性解的框架下,我们给出了新的验证定理,并得到有效策略(最优投资策略和最优再保险策略)的显式解和有效前沿.     

Optimal reinsurance–investment problem in a constant elasticity of variance stock market for jump‐diffusion risk model

In this paper, we consider the jump‐diffusion risk model with proportional reinsurance and stock price process following the constant elasticity of variance model. Compared with the geometric Brownian motion model, the advantage of the constant elasticity of variance model is that the volatility has correlation with the risky asset price, and thus, it can explain the empirical bias exhibited by the Black and Scholes model, such as volatility smile. Here, we study the optimal investment–reinsurance problem of maximizing the expected exponential utility of terminal wealth. By using techniques of stochastic control theory, we are able to derive the explicit expressions for the optimal strategy and value function. Numerical examples are presented to show the impact of model parameters on the optimal strategies. Copyright © 2011 John Wiley & Sons, Ltd.     

Robust optimal reinsurance–investment strategy with price jumps and correlated claims

This paper considers the robust optimal reinsurance–investment strategy selection problem with price jumps and correlated claims for an ambiguity-averse insurer (AAI). The correlated claims mean that future claims are correlated with historical claims, which is measured by an extrapolative bias. In our model, the AAI transfers part of the risk due to insurance claims via reinsurance and invests the surplus in a financial market consisting of a risk-free asset and a risky asset whose price is described by a jump–diffusion model. Under the criterion of maximizing the expected utility of terminal wealth, we obtain closed-form solutions for the robust optimal reinsurance–investment strategy and the corresponding value function by using the stochastic dynamic programming approach. In order to examine the influence of investment risk on the insurer’s investment behavior, we further study the time-consistent reinsurance–investment strategy under the mean–variance framework and also obtain the explicit solution. Furthermore, we examine the relationship among the optimal reinsurance–investment strategies of the AAI under three typical cases. A series of numerical experiments are carried out to illustrate how the robust optimal reinsurance–investment strategy varies with model parameters, and result analyses reveal some interesting phenomena and provide useful guidances for reinsurance and investment in reality.     

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.