The optimal barrier dividend problem under excess of loss reinsurance strategy has rarely been studied so far. We combine the risk factors such as market friction and terminal residual value with risk investment and risk control strategy, and study the resulting optimal investment-excess of loss reinsurance-barrier dividend problem. Based on the dynamic programming principle, we establish the Hamilton-Jacobi-Bellman equation, and obtain the explicit solutions for the optimal investment-excess of loss reinsurance strategy. The optimal dividend function is solved by the differential-integral method. The existence and uniqueness of the optimal dividend boundary is proved.

Considering the economic situation, we investigate the optimal asset allocation of defined contribution pension funds with random payouts after retirement under a modified criterion of quadratic loss. The HJB equation is derived adhering to the dynamic programming principle, and the time-consistent optimal investment strategy is designed based on the calculus theory. Finally, under two different risk attitudes, namely surplus preference and risk aversion, the impact of key parameters on the optimal investment strategy and the function of minimum loss at the initial moment is compared and analyzed, the economic significance is demonstrated, and the rationality of the model is verified.

We investigate a robust optimal reinsurance-investment problem for $ n $ insurers under multiple interactions, which arise from the insurance market, the financial market, the competition mechanism and the cooperation mechanism. Each insurer's surplus process is assumed to follow a diffusion model, which is an approximation of the classical Cramér-Lundberg model. Each insurer is allowed to purchase proportional reinsurance to reduce their claim risk. To reflect the first moment and second moment information on claims, we use the variance premium principle to calculate reinsurance premiums. To increase wealth, each insurer can invest in a financial market, which includes one risk-free asset and $ n $ correlated stocks. Each insurer wants to obtain the robust optimal reinsurance and investment strategy under the mean-variance criterion. By applying a stochastic control technique and dynamic programming approach, the extended Hamilton-Jacobi-Bellman (HJB) equation is established. Furthermore, we derive both the robust optimal reinsurance-investment strategy and the corresponding value function by solving the extended HJB equation. Finally, we present numerical experiments, which yield that competition and cooperation have an important influence on the insurer's decision-making.

在实际中,多个保险人之间经常存在竞争与合作.文章在竞争与合作统一框架下,研究了鲁棒最优再保险策略.每个保险人的盈余过程满足扩散逼近保险模型,n个保险人的索赔之间存在相依关系,每个保险人通过再保险减少索赔风险.文章主要的研究目标是,在最坏市场环境下,寻找最优均衡再保险策略最大化终端财富的均值同时最小化其方差.通过使用随机动态规划和随机控制理论,求得了鲁棒最优均衡再保险策略、最优市场策略和最优值函数的显式解,并从理论上探讨了最优策略的经济意义.最终,通过数值实验分析了竞争、合作、模糊厌恶和风险厌恶对鲁棒最优均衡再保险策略的影响.文章的研究结果可以有效地指导保险人的实践.

This paper studies the closed-loop equilibrium reinsurance-investment problem with insider information and default risk. The financial market consists of one risky asset, one defaultable bond, and one risk-free asset. The surplus process is governed by a jump-diffusion process. Two kinds of dependencies between the insurance market and the financial market are considered. In addition, the insurer has some extra claims information available from the beginning of the trading interval. The objective of the insurer is to choose a time-consistent reinsurance-investment strategy so as to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. Since this problem is time-inconsistent, using closed-loop control approach from the perspective of game theory, we establish the extended Hamilton–Jacobi–Bellman (HJB) equations for the postdefault case and the predefault case, respectively. Closed-form solutions for the closed-loop equilibrium reinsurance-investment strategy and the corresponding value function are obtained. Finally, we provide a series of numerical examples to illustrate the effects of insider information and other some important model parameters on the closed-loop equilibrium reinsurance and investment strategies. The result analyses reveal some interesting phenomena and provide useful guidances for reinsurance and investment in reality.

In reality, when facing a defined contribution (DC) pension fund investment problem, the fund manager may not have sufficient confidence in the reference model and rather considers some similar alternative models. In this paper, we investigate the robust equilibrium control-measure policy for an ambiguity-averse and risk-averse fund manger under the mean-variance (MV) criterion. The ambiguity aversion is introduced by adopting the model uncertainty robustness framework developed by Anderson. The risk aversion model is state-dependent, and takes a linear form of the current wealth level after contribution. Moreover, the fund manager faces stochastic labor income risk and allocates his wealth among a risk-free asset and a risky asset. We also propose two complicated ambiguity preference functions which are economically meaningful and facilitate analytical tractability. Due to the time-inconsistency of the resulting stochastic control problem, we attack it by using the game theoretical framework and the concept of subgame perfect Nash equilibrium. The extended Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations and the verification theorem for our problem are established. The explicit expressions for the robust equilibrium policy and the corresponding robust equilibrium value function are derived by stochastic control technique. In addition, we discuss two special cases of our model, which shows that our results extend some existing works in the literature. Finally, some numerical experiments are conducted to demonstrate the effects of model parameters on our robust equilibrium policy.

We propose a new interaction mechanism for n≥2 competitive insurers, under which we investigate the time-consistent equilibrium reinsurance–investment strategy for n insurers. Each insurer can purchase reinsurance for reducing the claim risk and invest in the financial market for increasing his wealth. We consider a general investment framework which includes typical diversified and concentrated investment patterns in the literature. The objective of each insurer is to find a time-consistent equilibrium reinsurance–investment strategy so as to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. By using the stochastic control technique, we obtain the explicit time-consistent equilibrium reinsurance–investment strategies and corresponding equilibrium value functions. Through analyzing the general investment strategy, we find the criterion of whether the insurer should invest in a specific risky asset. Furthermore, we examine theoretically and numerically how the price parameters of risky assets affect the concrete investment pattern. We find that when the appreciation rate of one risky asset exceeds some times as those of other risky assets, the insurer will only invest in this asset; otherwise, the insurer will adopt the diversified investment pattern. We also numerically examine the influences of the number of insurers and the surplus jump on the time-consistent equilibrium reinsurance strategy.

A robust time-consistent optimal investment strategy selection problem under inflation influence is investigated in this article. The investor may invest his wealth in a financial market, with the aim of increasing wealth. The financial market includes one risk-free asset, one risky asset, and one inflation-indexed bond. The price process of the risky asset is governed by a constant elasticity of variance (CEV) model. The investor is ambiguity-averse; he doubts about the model setting under the original probability measure. To dispel this concern, he seeks a set of alternative probability measures, which are absolutely continuous to the original probability measure. The objective of the investor is to seek a time-consistent strategy so as to maximize his expected terminal wealth meanwhile minimizing his variance of the terminal wealth in the worst-case scenario. By using the stochastic optimal control technique, we derive closed-form solutions for the optimal time-consistent investment strategy, the probability scenario, and the value function. Finally, the influences of model parameters on the optimal investment strategy and utility loss function are examined through numerical experiments.

本文研究了一个保险公司经营n类相依保险业务下,最优时间一致的再保险和投资问题.为了减少理赔风险,保险公司可以购买再保险;为了增加财富保险公司可以在金融市场上投资.金融市场由一个无风险资产和n个相依的风险资产组成,风险资产的价格满足扩散过程.然后,利用随机分析理论,我们建立了保险公司的财富过程.我们的主要目标是,寻找最优时间一致的再保险和投资策略最大化终值财富的均值同时最小化终值财富的方差.通过使用随机控制和随机动态规划技术,我们建立了推广的Hamilton-Jacob-Bellman(HJB)方程.进而,通过求解推广的HJB方程,我们得到了最优时间一致的再保险和投资策略以及相应值函数的显式解.最终,通过数值实验解释了模型参数对最优时间一致的再保险和投资策略的影响.

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.

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.

考虑了通货膨胀影响下时间一致的投资策略选择问题,运用随机控制的方法得到了时间一致的最优投资策略和相应值函数的显式解,并通过数值算例讨论了模型参数对时间一致最优投资策略的影响.

本文在通胀和负债影响下研究时间一致的投资策略选择问题.通胀和负债满足扩散过程,风险资产的价格用Levy过程刻画,并且考虑通胀、负债与风险资产之间的相关性.以最大化终止盈余的均值,同时最小化终止盈余的方差为目标,应用随机动态规划的方法研究该问题,得到最优时间一致投资策略和值函数的显式解.最后,通过数值计算,解释了通胀、负债对最优时间一致投资策略的影响.

This paper investigate a stochastic differential games for DC (defined contribution plans) pension under Vasicek stochastic interest rate. The finance market as the hypothetical counterpart, the investor as pension the leader of game. Our goal is through the game between pension plan investor and financial market, obtain optimal strategies to maximizes the expected utility of the terminal wealth. Under power utility function, by using stochastic control theory, we obtain closed-form solutions for the value function as well as the strategies. Finally, explain the research results in the economic sense, and though numerical calculation given the influence of some parameters on the optimal strategies.

Under inflation influence,an optimal time-consistent strategy selection problem for claims dependent risk model is studied.The two claim number processes are correlated by a common Poisson process.The insurer can purchase reinsurance for reducing claims and invest its surplus in finance market for increasing wealth.In the process of investment,the effect of inflation is taken into account,and the effect of inflation is achieved through the conversion of the risk asset to the inflation rate.The objective of the insurer is to choose an optimal time-consistent reinsurance-investment strategy so as to maximize the expected terminal surplus while minimizing the variance of the terminal surplus.Since this problem is time-inconsistent,it is studied by placing the problem within a game theoretic framework.Applying Hamilton-Jacobi-Bellman dynamic programming approach,closed form solutions for the optimal reinsurance-investment strategy and the corresponding value functions are obtained.Finally,the influence of some insurance market model parameters on optimal reinsurance strategy is explained by numerical calculation,and the influence of financial market model parameters and inflation model parameters on optimal investment strategy are also given.Through this study,it can guide investors to make reasonable investment under the influence of inflation,so that their wealth is the largest and the smallest risk.

Taking an example of three people,stochastic differential game between multiple decision makers is studied.Two of them are the mutual cooperation investors and the last one is the two investors "virtual" opponents-the financial markets.Two kinds of stochastic differential games are studied,one is the game based on utility and the other one is based on the mean-variance criterion.For the first case,the goal of two investors is to make the expected utility of the final value wealth maximum,the financial market goal is to minimize the expected utility.For the second case,the two investor target is to minimize the variance of the terminal value of wealth with a given final expected value.For financial markets the goal is to make the variance of financial markets maximum.By applying stochastic control theory,we obtained the optimal investment strategy,the optimal market strategy and the explicit solution of the optimal value function two game problems.Our result can guide the choice of appropriate investment strategy in the mutual cooperation of the two investors in the financial market to maximize the expected utility of the final wealth,or to minimize risk for wealth for themselves.

Zero-sum stochastic differential games between insurance company and financial market are considered.The goal is to obtain optimal strategies to maximize the expected utility of the terminal wealth by the game between insurance company and financial market.Under power utility function,by using stochastic control theory,the closed-form solutions for the value function as well as the strategies is obtained.Finally,the research results are explained in the economic sense and the influence of some parameters on the optimal strategies is given through numerical calculation.

An optimal reinsurance and investment strategy selection in a risk model with two dependent classes of insurance business is considered.The objective of the insurer is to choose an optimal timeconsistent reinsurance-investment strategy so as to maximize the expected terminal surplus while minimizing the variance of the terminal surplus.By using the dynamic planning approach,closed-form solutions for the optimal reinsurance and investment strategies and the corresponding value functions are obtained.Numerical examples and theoretical analysis are also provided to illustrate how the optimal reinsurance and investment strategies changes when some model parameters vary.

Under mean-variance criterion,this paper studies optimal reinsurance and investment for insurance company.The surplus of the insurance company satisfy Cramer-Lundberg risk model.Reinsurance and investment in finance market are adopted by the insurance company to reduce risk and increase profit.The risky asset price is describe by a Ornstein-Uhlenbeck (O-U) model.The studies aim are obtain closed form of optimal reinsurance and investment strategies and efficient frontier.Through linear-quadratic (LQ) control theory and It5 Formula,we solve the problem.Through this paper studies not only enrich and develop the strategy selection problem but also have some significance for insurance companies when it reinsurance and investment.

This paper studies the problem of stochastic differential game between investor and market with transaction costs and liability.Our main goal is to find an optimal investment and market policy which maximizes the expected exponential utility of the terminal wealth.By applying linear quadratic control theory,closed-form solutions for the value function as well as the optimal investment and market policy are obtained.Finally,further explanations on the economy are given by analyzing the obtained results.