Optimal risk control and dividend policies under excess of loss reinsurance

We study the optimal reinsurance policy and dividend distribution of an insurance company under excess of loss reinsurance. The objective of the insurer is to maximize the expected discounted dividends. We suppose that in the absence of dividend distribution, the reserve process of the insurance company follows a compound Poisson process. We first prove existence and uniqueness results for this optimization problem by using singular stochastic control methods and the theory of viscosity solutions. We then compute the optimal strategy of reinsurance, the optimal dividend strategy and the value function by solving the associated integro-differential Hamilton–Jacobi–Bellman Variational Inequality numerically.     

Classical and Impulse Control for the Optimization of Dividend and Proportional Reinsurance Policies with Regime Switching

We consider the optimal proportional reinsurance and dividend strategy. The surplus process is modeled by the classical compound Poisson risk model with regime switching. Considering a class of utility functions, the object of the insurer is to select the reinsurance and dividend strategy that maximizes the expected total discounted utility of the shareholders until ruin. By adapting the techniques and methods of stochastic control, we study the quasi-variational inequality for this classical and impulse control problem and establish a verification theorem. We show that the optimal value function is characterized as the unique viscosity solution of the corresponding quasi-variational inequality.     

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.     

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.     

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.     

方差保费准则下的最优脉冲控制

本文研究保险公司在有再保险控制下的最优脉冲分红问题. 对保险公司的理赔损失, 假定有两家再保险公司参与分保, 且保险公司与两家再保险公司采取不同参数下的方差保费准则. 进一步, 假定保险公司有股东红利分配, 且每次分红有固定交易费和比例税收, 即脉冲分红. 在扩散逼近模型下, 本文应用随机动态规划方法研究破产前的最大期望折现分红, 给出值函数的解析表达式, 进而获得最优再保险策略和分红策略的具体形式.     

Viscosity solution and impulse control of the diffusion model with reinsurance and fixed transaction costs

We consider an optimal impulse control problem on reinsurance, dividend and reinvestment of an insurance company. To close reality, we add fixed and proportional transaction costs to this problem. The value of the company is associated with expected present value of net dividends pay out minus the net reinvestment capitals until ruin time. We focus on non-cheap proportional reinsurance. We prove that the value function is a unique solution to associated Hamilton–Jacobi–Bellman equation, and establish the regularity property of the viscosity solution under a weak assumption. We solve the non-uniformly elliptic equation associated with the impulse control problem. Finally, we derive the value function and the optimal strategy of the control problem.     

Optimal risk and dividend control of an insurance company with exponential premium principle and liquidation value

Assume that an insurer can control it’s surplus by paying dividends, purchasing reinsurance and injecting capital. The exponential premium principle is used when pricing insurance contract instead of the expected value principle. Under the objective of maximizing the company’s value, we identify the optimal strategies with liquidation value and transaction costs. The results illustrate that the insurer should buy less reinsurance when the surplus increases, capital injection should be considered if and only if the transaction costs and the liquidation value are relatively low, dividends are paid according to barrier strategy if the dividend rate is unrestricted or threshold strategy if the dividend rate is bounded.     

Dynamic risk-sharing game and reinsurance contract design

This paper studies the optimal risk-sharing between an insurer and a reinsurer. The insurer purchases reinsurance for risk-control and decides her retention level with an objective to minimize her ruin probability. The reinsurer has control over the reinsurance price and aims to maximize her expected discounted profits up to the time when the insurer goes bankrupt. In a stochastic differential game-theoretic framework, we determine the insurer’s optimal reinsurance strategy and specify the reinsurance contract by solving a system of coupled Hamilton–Jacobi–Bellman equations. We obtain explicit solutions for the game problem when both the insurance and the reinsurance premiums are calculated according to the standard-deviation principle or the expected value principle, respectively. Our results show that, depending on the model parameters, the reinsurance contract is either provided with a peak price when the insurer has sufficient cash reserve and with a minimum price when otherwise, or is always provided with a peak price. We also perform some numerical analyses and provide economic interpretations for the results.     

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.     

Stochastic Stackelberg differential reinsurance games under time-inconsistent mean–variance framework

We study optimal reinsurance in the framework of stochastic Stackelberg differential game, in which an insurer and a reinsurer are the two players, and more specifically are considered as the follower and the leader of the Stackelberg game, respectively. An optimal reinsurance policy is determined by the Stackelberg equilibrium of the game, consisting of an optimal reinsurance strategy chosen by the insurer and an optimal reinsurance premium strategy by the reinsurer. Both the insurer and the reinsurer aim to maximize their respective mean–variance cost functionals. To overcome the time-inconsistency issue in the game, we formulate the optimization problem of each player as an embedded game and solve it via a corresponding extended Hamilton–Jacobi–Bellman equation. It is found that the Stackelberg equilibrium can be achieved by the pair of a variance reinsurance premium principle and a proportional reinsurance treaty, or that of an expected value reinsurance premium principle and an excess-of-loss reinsurance treaty. Moreover, the former optimal reinsurance policy is determined by a unique, model-free Stackelberg equilibrium; the latter one, though exists, may be non-unique and model-dependent, and depend on the tail behavior of the claim-size distribution to be more specific. Our numerical analysis provides further support for necessity of integrating the insurer and the reinsurer into a unified framework. In this regard, the stochastic Stackelberg differential reinsurance game proposed in this paper is a good candidate to achieve this goal.     

Optimal investment and reinsurance for an insurer under Markov-modulated financial market

This study examines optimal investment and reinsurance policies for an insurer with the classical surplus process. It assumes that the financial market is driven by a drifted Brownian motion with coefficients modulated by an external Markov process specified by the solution to a stochastic differential equation. The goal of the insurer is to maximize the expected terminal utility. This paper derives the Hamilton–Jacobi–Bellman (HJB) equation associated with the control problem using a dynamic programming method. When the insurer admits an exponential utility function, we prove that there exists a unique and smooth solution to the HJB equation. We derive the explicit optimal investment policy by solving the HJB equation. We can also find that the optimal reinsurance policy optimizes a deterministic function. We also obtain the upper bound for ruin probability in finite time for the insurer when the insurer adopts optimal policies.     

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.     

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.     

扩散风险模型下再保险和投资对红利的影响

对扩散风险模型,研究了比例再保险和投资对红利的影响．在常数边界分红策略下,得到了使得期望贴现红利最大的最优比例再保险和投资策略的显示表达式,并得到最大期望贴现红利的显示表达式．最后,通过数值计算得到了再保险和投资对期望红利的影响,以及最优投资策略与各参数之间的关系．     

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.     

Optimality of excess-loss reinsurance under a mean–variance criterion

In this paper, we study an insurer’s reinsurance–investment problem under a mean–variance criterion. We show that excess-loss is the unique equilibrium reinsurance strategy under a spectrally negative Lévy insurance model when the reinsurance premium is computed according to the expected value premium principle. Furthermore, we obtain the explicit equilibrium reinsurance–investment strategy by solving the extended Hamilton–Jacobi–Bellman equation.     

最小化破产概率的保险人鲁棒投资再保险策略研究

在模型不确定条件下,研究以破产概率最小化为目标的模糊厌恶型保险公司的最优投资再保险问题. 假设保险公司可投资于一种风险资产,也可购买比例再保险. 分别考虑风险资产的价格过程服从随机波动率模型和非随机波动率模型的两种情况,根据动态规划原理建立相应的HJB方程,得到保险公司的最优鲁棒投资再保险策略和价值函数的解析解. 最后,通过数值模拟分析了各模型参数对最优策略和价值函数的影响.     

The optimal policy for insurance company under consideration of internal competition and the time value of ruin

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.