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.     

Optimal proportional reinsurance with common shock dependence

In this paper, we consider the optimal proportional reinsurance strategy in a risk model with multiple dependent classes of insurance business, which extends the work of Liang and Yuen (2014) to the case with the reinsurance premium calculated under the expected value principle and to the model with two or more classes of dependent risks. Under the criterion of maximizing the expected exponential utility, closed-form expressions for the optimal strategies and value function are derived not only for the compound Poisson risk model but also for the diffusion approximation risk model. In particular, we find that the optimal reinsurance strategies under the expected value premium principle are very different from those under the variance premium principle in the diffusion risk model. The former depends not only on the safety loading, time and interest rate, but also on the claim size distributions and the counting processes, while the latter depends only on the safety loading, time and interest rate. Finally, numerical examples are presented to show the impact of model parameters on the optimal strategies.     

The American Foreign Exchange Option in Time-Dependent One-Dimensional Diffusion Model for Exchange Rate

The classical Garman-Kohlhagen model for the currency exchange assumes that the domestic and foreign currency risk-free interest rates are constant and the exchange rate follows a log-normal diffusion process. In this paper we consider the general case, when exchange rate evolves according to arbitrary one-dimensional diffusion process with local volatility that is the function of time and the current exchange rate and where the domestic and foreign currency risk-free interest rates may be arbitrary continuous functions of time. First non-trivial problem we encounter in time-dependent case is the continuity in time argument of the value function of the American put option and the regularity properties of the optimal exercise boundary. We establish these properties based on systematic use of the monotonicity in volatility for the value functions of the American as well as European options with convex payoffs together with the Dynamic Programming Principle and we obtain certain type of comparison result for the value functions and corresponding exercise boundaries for the American puts with different strikes, maturities and volatilities. Starting from the latter fact that the optimal exercise boundary curve is left continuous with right-hand limits we give a mathematically rigorous and transparent derivation of the significant early exercise premium representation for the value function of the American foreign exchange put option as the sum of the European put option value function and the early exercise premium. The proof essentially relies on the particular property of the stochastic integral with respect to arbitrary continuous semimartingale over the predictable subsets of its zeros. We derive from the latter the nonlinear integral equation for the optimal exercise boundary which can be studied by numerical methods.     

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.     

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.     

广义负相依一般风险模型中有限时破产概率的估计及数值模拟

本文研究了一类带利率的重尾相依风险模型, 其中索赔额是一列上广义负相依随机变量, 索赔到达过程是一般的非负整值过程, 并且独立于索赔额序列, 保费收入过程是一个一般的非负非降随机过程. 我们考虑了两种情况, 其一是索赔额、索赔到达过程及保费收入过程相互独立, 其二是累积折现保费收入总量的尾概率可以被索赔额的尾概率高阶控制, 得到了保险公司有限时破产概率的渐近估计,并且给出了相应的数值模拟, 验证了理论结果的合理性.     

Optimum excess-loss reinsurance: a dynamic framework

The purpose of this article is to consider a two firms excess-loss reinsurance problem. The first firm is defined as the direct underwriter while the second firm is the reinsurer. As in the classical model of collective risk theory it is assumed that premium payments are received deterministically from policyholders at a constant rate, while the claim process is determined by a compound Poisson process. The objective of the underwriter is to maximize the expected present value of the long run terminal wealth (investments plus cash) of the firm by selecting an appropriate excess-loss coverage strategy, while the reinsurer seeks to maximize its total expected discounted profit by selecting an optimal loading factor. Since both firms' policies are interdependent we define an insurance game, solved by employing a Stackelberg solution concept. A diffusion approximation is used in order to obtain tractable results for a general claim size distribution. Finally, an example is presented illustrating computational procedures.     

Optimal investment policy of an insurance firm

In this paper we consider an investment problem by an insurance firm. As in the classical model of collective risk, it is assumed that premium payments are received deterministically from policyholders at a constant rate, while the claim process is determined by a compound Poisson process. We introduce a conversion mechanism of funds from cash into investments and vice versa. Contrary to the conventional collective risk model we do not assume a ruin barrier. Instead we introduce conversion costs to account for the problems implicit in reaching the zero boundary. The objective of the firm is to maximize its net profit by selecting an appropriate investment strategy. A diffusion approximation is suggested in order to obtain tractable results for a general claim size distribution.     

Constrained investment–reinsurance optimization with regime switching under variance premium principle

This paper studies optimal investment and reinsurance problems for an insurer under regime-switching models. Two types of risk models are considered, the first being a Markov-modulated diffusion approximation risk model and the second being a Markov-modulated classical risk model. The insurer can invest in a risk-free bond and a risky asset, where the underlying models for investment assets are modulated by a continuous-time, finite-state, observable Markov chain. The insurer can also purchase proportional reinsurance to reduce the exposure to insurance risk. The variance principle is adopted to calculate the reinsurance premium, and Markov-modulated constraints on both investment and reinsurance strategies are considered. Explicit expressions for the optimal strategies and value functions are derived by solving the corresponding regime-switching Hamilton–Jacobi–Bellman equations. Numerical examples for optimal solutions in the Markov-modulated diffusion approximation model are provided to illustrate our results.     

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.     

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.     

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.     

On the Gerber-Shiu Function and Optimal Dividend Strategy for a Thinning Risk Model

In this paper, the risk model under constant dividend barrier strategy is studied, in which the premium income follows a compound Poisson process and the arrival of the claims is a p-thinning process of the premium arrival process. The integral equations with boundary conditions for the expected discounted aggregate dividend payments and the expected discounted penalty function until ruin are derived. In addition, the explicit expressions for the Laplace transform of the ruin time and the expected aggregate discounted dividend payments until ruin are given when the individual stochastic premium amount and claim amount are exponentially distributed. Finally, the optimal barrier is presented under the condition of maximizing the expectation of the difference between discounted aggregate dividends until ruin and the deficit at ruin.     

相关双边跳扩散模型的期望折现罚金函数

带干扰的经典风险模型,其干扰项可被解释为未来的总理赔量,保费收入以及未来投资收益的不确定性,用双指数跳扩散过程来描述这些不确定性,考虑双边跳扩散模型的期望折现罚金函数,给出其所满足的积分微分方程,并给出破产时间和破产时公司现值的联合拉普拉斯变换的显式表达公式.     

索赔为稀疏过程的双复合 Poisson风险模型

研究了一类风险过程,其中保费收入为复合Poisson过程,而描述索赔发生的计数过程为保单到达过程的p-稀疏过程.给出了生存概率满足的积分方程及其在指数分布下的具体表达式,得到了破产概率满足的Lundberg不等式、最终破产概率及有限时间内破产概率的一个上界和生存概率的积分-微分方程,且通过数值例子,分析了初始准备金、保费收入、索赔支付及保单的平均索赔比例对保险公司破产概率的影响.     

Optimal buffer size and dynamic rate control for a queueing system with impatient customers in heavy traffic

We address a rate control problem associated with a single server Markovian queueing system with customer abandonment in heavy traffic. The controller can choose a buffer size for the queueing system and also can dynamically control the service rate (equivalently the arrival rate) depending on the current state of the system. An infinite horizon cost minimization problem is considered here. The cost function includes a penalty for each rejected customer, a control cost related to the adjustment of the service rate and a penalty for each abandoning customer. We obtain an explicit optimal strategy for the limiting diffusion control problem (the Brownian control problem or BCP) which consists of a threshold-type optimal rejection process and a feedback-type optimal drift control. This solution is then used to construct an asymptotically optimal control policy, i.e. an optimal buffer size and an optimal service rate for the queueing system in heavy traffic. The properties of generalized regulator maps and weak convergence techniques are employed to prove the asymptotic optimality of this policy. In addition, we identify the parameter regimes where the infinite buffer size is optimal.     

考虑污染和种内关系的种群扩散最优控制模型

针对污染和种内关系均影响细菌种群扩散这一管理生态学问题,本文建立了基于非线性拟抛物方程的最优控制模型,将外界环境向细菌种群输入的毒素率作为控制变量,运用控制理论和方法探讨污染和种内关系双重影响下种群扩散系统的最优控制问题。利用Schauder不动点定理证明了该种群扩散系统的适定性;同时,通过建立新的Carleman型估计,给出了容许控制和最优控制的存在性。最后,通过数值算例分析了理论推导的结果,在算例中都找到一对时间最优控制,验证了种群扩散系统最优控制模型的有效性。该研究结果对现代传染病预防具有借鉴意义,也为有效控制瘟疫的爆发和流行提供理论参考。     

Global optimization of polynomial-expressed nonlinear optimal control problems with semidefinite programming relaxation

In this paper, we propose a new deterministic global optimization method for solving nonlinear optimal control problems in which the constraint conditions of differential equations and the performance index are expressed as polynomials of the state and control functions. The nonlinear optimal control problem is transformed into a relaxed optimal control problem with linear constraint conditions of differential equations, a linear performance index, and a matrix inequality condition with semidefinite programming relaxation. In the process of introducing the relaxed optimal control problem, we discuss the duality theory of optimal control problems, polynomial expression of the approximated value function, and sum-of-squares representation of a non-negative polynomial. By solving the relaxed optimal control problem, we can obtain the approximated global optimal solutions of the control and state functions based on the degree of relaxation. Finally, the proposed global optimization method is explained, and its efficacy is proved using an example of its application.     

最优投资组合模型研究

本文研究了在完备金融市场上 ,投资者最优投资组合的随机模型。在模型参数为常系数 ,效用函数为 (0 ,T],B[0 ,T])上的有界可测函数的情形下 ,得出其最大效用值函数是随机控制问题对应的 HJB方程的平滑解 ;最优策略被证明是存在的 ,并用反馈形式给出了最优投资组合策略。     

Optimal pricing and production in an inventory model

This paper deals with the problem of simultaneously determining the optimal price policy and production rate over a given planning horizon. For nonlinear demand functions and convex inventory and shortage cost functions the optimal solution paths are derived by using optimal control theory. The treatment of linear nonsmooth cost functions requires the use of a generalized maximum principle. The solution method is a phase portrait analysis providing insight into the optimal pricing and production policies as well as the resulting inventory paths. Moreover, it is shown that in the case of nonsmooth piecewise linear cost functions the equilibrium is approached within finite time although the model is nonlinear in the control variables. Finally it is illustrated that exogenous fluctuations in the demand rate (seasonal demand pattern) amount to cyclical optimal solutions.