Optimal proportional reinsurance model with transaction costs

Based on the mixed control strategy (regular control and impulse dividend control strategy), we formulate a proportional reinsurance model with transaction costs. For getting the maximal return function and associated mixed control strategy, using Itô calculus and classical mixed control theory, we derive the quasi-variational inequality solution to this optimal problem. Furthermore, we obtain its closed forms under some assumptions.     

比例再保险模型的脉冲与正则 最优控制策略

首次在比例再保险模型中引入脉冲控制过程,为获得最优回报函数,不但给出了该函数所应满足的充分性定理,而且得出了该函数的具体解析式及相应的最优脉冲与正则控制策略.     

Maximum Principle of Optimal Control of the Primitive Equations of the Ocean with Two Point Boundary State Constraint

We consider a class of stochastic impulse control problems of general stochastic processes i.e. not necessarily Markovian. Under fairly general conditions we establish existence of an optimal impulse control. We also prove existence of combined optimal stochastic and impulse control of a fairly general class of diffusions with random coefficients. Unlike, in the Markovian framework, we cannot apply quasi-variational inequalities techniques. We rather derive the main results using techniques involving reflected BSDEs and the Snell envelope.     

An investment model with switching costs and the option to abandon

We develop a complete analysis of a general entry–exit–scrapping model. In particular, we consider an investment project that operates within a random environment and yields a payoff rate that is a function of a stochastic economic indicator such as the price of or the demand for the project’s output commodity. We assume that the investment project can operate in two modes, an “open” one and a “closed” one. The transitions from one operating mode to the other one are costly and immediate, and form a sequence of decisions made by the project’s management. We also assume that the project can be permanently abandoned at a discretionary time and at a constant sunk cost. The objective of the project’s management is to maximise the expected discounted payoff resulting from the project’s management over all switching and abandonment strategies. We derive the explicit solution to this stochastic control problem that involves impulse control as well as discretionary stopping. It turns out that this has a rather rich structure and the optimal strategy can take eight qualitatively different forms, depending on the problems data.     

保险人投资策略与破产的关系

本文主要考虑带投资收益的风险模型,在该模型下保险人可以根据盈余投资,投资的数量为时间t的函数,我们得到保险人投资策略与破产概率与t时刻所满足的积分-微分方程.     

Analysis of survivorship life insurance portfolios with stochastic rates of return

A general portfolio of survivorship life insurance contracts is studied in a stochastic rate of return environment with a dependent mortality model. Two methods are used to derive the first two moments of the prospective loss random variable. The first one is based on the individual loss random variables while the second one studies annual stochastic cash flows. The distribution function of the present value of future losses at a given valuation time is derived. For illustrative purposes, an AR(1) process is used to model the stochastic rates of return, and the future lifetimes of a couple are assumed to follow a copula model. The effects of the mortality dependence, the portfolio size and the policy type, as well as the impact of investment strategies on the riskiness of portfolios of survivorship life insurance policies are analyzed by means of moments and probability distributions.     

一类半鞅状态的平稳型脉冲随机控制

本文提出了一类新的随机控制模型,这类模型不但在费用结构上推广了此前的平稳型脉冲随机控制,而且首次将一类半鞅引入脉冲控制模型的状态结构从而推广了相应的状态过程.通过对一类相当复杂的变分方程问题的研究并利用其有关结论,我们证明了新模型最佳控制的存在性并刻划出其结构.     

资产组合,随机经济增长和政府政策的影响

本文在一个连续时间的经济增长模型中考虑政府政策、投资策略和个体福利 .在给定的生产技术 ,偏好和随机冲击的假设下 ,本文得到了持有资产税后期望回报、随机经济增长率、消费财富比、资产组合份额和各种资产实际回报率的显式解 .     

Service rate control of closed Jackson networks from game theoretic perspective

Game theoretic analysis of queueing systems is an important research direction of queueing theory. In this paper, we study the service rate control problem of closed Jackson networks from a game theoretic perspective. The payoff function consists of a holding cost and an operating cost. Each server optimizes its service rate control strategy to maximize its own average payoff. We formulate this problem as a non-cooperative stochastic game with multiple players. By utilizing the problem structure of closed Jackson networks, we derive a difference equation which quantifies the performance difference under any two different strategies. We prove that no matter what strategies the other servers adopt, the best response of a server is to choose its service rates on the boundary. Thus, we can limit the search of equilibrium strategy profiles from a multidimensional continuous polyhedron to the set of its vertex. We further develop an iterative algorithm to find the Nash equilibrium. Moreover, we derive the social optimum of this problem, which is compared with the equilibrium using the price of anarchy. The bounds of the price of anarchy of this problem are also obtained. Finally, simulation experiments are conducted to demonstrate the main idea of this paper.     

考虑收益率及破产补偿函数的奇异最优随机控制模型

通过在目标结构中引入收益率及破产补偿函数,建立了一非对称型最优奇异随机控制模型.利用随机积分及最优控制理论,得出了最大回报函数的显式解及相应的最优控制策略.     

Dynamic programming for multidimensional stochastic control problems

In this paper we study a general multidimensional diffusion-type stochastic control problem. Our model contains the usual regular control problem, singular control problem and impulse control problem as special cases. Using a unified treatment of dynamic programming, we show that the value function of the problem is a viscosity solution of certain Hamilton-Jacobi-Bellman (HJB) quasivariational inequality. The uniqueness of such a quasi-variational inequality is proved. Supported in part by USA Office of Naval Research grant #N00014-96-1-0262. Supported in part by the NSFC Grant #79790130, the National Distinguished Youth Science Foundation of China Grant #19725106 and the Chinese Education Ministry Science Foundation.     

Optimal risk and liquidity management with costly refinancing opportunities

In this paper we study risk and liquidity management decisions within an insurance firm. Risk management corresponds to decisions regarding proportional reinsurance, whereas liquidity management has two components: distribution of dividends and costly equity issuance. Contingent on whether proportional or fixed costs of reinvestment are considered, singular stochastic control or stochastic impulse control techniques are used to seek strategies that maximize the firm value. We find that, in a proportional-costs setting, the optimal strategies are always mixed in terms of risk management and refinancing. In contrast, when fixed issuance costs are too high relative to the firm’s profitability, optimal management does not involve refinancing. We provide analytical specifications of the optimal strategies, as well as a qualitative analysis of the interaction between refinancing and risk management.     

Optimal control problems with no turning back

In this paper we consider the maximization of a payoff functional subject to a differential equality contraint over the class of monotonically increasing functions with values in [0, 1]. We will show that an optimal control exists, derive the system of inequalities (similar to a quasi-variational inequality) that the value function satisfies and derive various properties of the value function sufficient to characterize it. Furthermore, we derive a perturbation result using the theory of Lipschitz controls. Finally, we also consider the case when the control functions are of bounded total variation and relate the problems considered herein to the impulse control problem of Bensoussan-Lions.     

Lipschitz continuous dynamic programming with discount

We show that if the return function, the technological constraints and the transition function of a standard problem of stochastic dynamic programming with discount satisfy Lipschitz regularity assumptions, then the value function is Lipschitz regular.     

Comparison theorems for stochastic evolution equations

This paper establishes an anticipating stochastic differential equation of parabolic type for the expectation of the solution of a stochastic differential equation conditioned on complete knowledge of the path of one of its components. Conversely, it is shown that any appropriately regular solution of this stochastic p.d.e. must be given by the conditional expectation. These results generalize the connection, known as the Feynman-Kac formula, between parabolic equations and expectations of functions of a diffusion. As an application, we derive an equation for the unnormalized smoothing law of a filtering problem with observation feedback.     

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.     

跳扩散模型下基金平衡管理的最优脉冲控制

在基金市值波动服从跳扩散过程, 基金持有的罚金成本为当前基金水平的二次函数及存在交易费的假设下研究了无穷时域的基金平衡管理的最小成本模型. 利用随机最优脉冲控制的拟变分不等式理论建立了判定定理,得到了最优脉冲控制策略的存在性, 同时通过构造方法给出了解的数学结构形式.     

基于鲁棒二阶随机占优的投资组合优化模型研究

本文主要考虑一类经典的含有二阶随机占优约束的投资组合优化问题,其目标为最大化期望收益,同时利用二阶随机占优约束度量风险,满足期望收益二阶随机占优预定的参考目标收益。与传统的二阶随机占优投资组合优化模型不同,本文考虑不确定的投资收益率,并未知其精确的概率分布,但属于某一不确定集合,建立鲁棒二阶随机占优投资组合优化模型,借助鲁棒优化理论,推导出对应的鲁棒等价问题。最后,采用S&P 500股票市场的实际数据,对模型进行不同训练样本规模和不确定集合下的最优投资组合的权重、样本内和样本外不确定参数对期望收益的影响的分析。结果表明,投资收益率在最新的历史数据规模下得出的投资策略,能够获得较高的样本外期望收益,对未来投资更具参考意义。在保证样本内解的最优性的同时,也能取得较高的样本外期望收益和随机占优约束被满足的可行性。     

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.     

Asymptotics in a time-dependent renewal risk model with stochastic return

In this paper we study the asymptotic tail behavior for a non-standard renewal risk model with a dependence structure and stochastic return. An insurance company is allowed to invest in financial assets such as risk-free bonds and risky stocks, and the price process of its portfolio is described by a geometric Lévy process. By restricting the claim-size distribution to the class of extended regular variation (ERV) and imposing a constraint on the Lévy process in terms of its Laplace exponent, we obtain for the tail probability of the stochastic present value of aggregate claims a precise asymptotic formula, which holds uniformly for all time horizons. We further prove that the corresponding ruin probability also satisfies the same asymptotic formula.