Transferring a system with an unknown disturbance under the optimal control to a state of dynamic balance and to an ε-vicinity of the final state

A problem of transferring a linear system to a state of dynamic balance under a simultaneous action of an unknown disturbance and time-optimal control is considered. The optimal control is calculated along the phase trajectory, and it is periodically updated for discrete phase coordinate values. It is proved that the phase trajectory comes to the dynamic equilibrium point and makes undamped periodic motions (a stable limit cycle). The location of the dynamic equilibrium point and the limit cycle form are considered as functions of various parameters. With the disturbance calculated in the process of control, the accuracy of transferring to the required final state increases. A method for estimating attainable accuracy is presented. Results of simulation and numerical calculations are given.     

Maintenance outsourcing coordination with risk-averse contractors

Under a maintenance outsourcing contract, an external contractor receives a fixed payment from a manufacturer for periodically performing preventive maintenance and for performing minimal repairs whenever process failures occur. If the contractor’s maintenance policy results in a process uptime above a target level specified in the contract, the contractor receives a bonus payment based on the difference between the uptime and the target levels. We study the optimal designs of maintenance outsourcing contracts for achieving channel coordination when the contractor is risk averse towards uncertain repair costs caused by process failures. We find cases in which channel coordination cannot be achieved because of the contractor’s risk preference. Furthermore, the contractor’s risk preference may make channel coordination more difficult or easier, depending on the parameters considered in the model.     

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.     

An optimal investment strategy for a stream of liabilities generated by a step process in a financial market driven by a Lévy process

In this paper we investigate an asset–liability management problem for a stream of liabilities written on liquid traded assets and non-traded sources of risk. We assume that the financial market consists of a risk-free asset and a risky asset which follows a geometric Lévy process. The non-tradeable factor (insurance risk or default risk) is driven by a step process with a stochastic intensity. Our framework allows us to consider financial risk, systematic and unsystematic insurance loss risk (including longevity risk), together with possible dependencies between them. An optimal investment strategy is derived by solving a quadratic optimization problem with a terminal objective and a running cost penalizing deviations of the insurer’s wealth from a specified profit-solvency target. Techniques of backward stochastic differential equations and the weak property of predictable representation are applied to obtain the optimal asset allocation.     

Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process

In this paper, we study the optimal investment and proportional reinsurance strategy when an insurance company wishes to maximize the expected exponential utility of the terminal wealth. It is assumed that the instantaneous rate of investment return follows an Ornstein-Uhlenbeck process. Using stochastic control theory and Hamilton-Jacobi-Bellman equations, explicit expressions for the optimal strategy and value function are derived not only for the compound Poisson risk model but also for the Brownian motion risk model. Further, we investigate the partially observable optimization problem, and also obtain explicit expressions for the optimal results.     

Precommitment and equilibrium investment strategies for defined contribution pension plans under a jump–diffusion model

In this paper, we study an optimal investment problem under the mean–variance criterion for defined contribution pension plans during the accumulation phase. To protect the rights of a plan member who dies before retirement, a clause on the return of premiums for the plan member is adopted. We assume that the manager of the pension plan is allowed to invest the premiums in a financial market, which consists of one risk-free asset and one risky asset whose price process is modeled by a jump–diffusion process. The precommitment strategy and the corresponding value function are obtained using the stochastic dynamic programming approach. Under the framework of game theory and the assumption that the manager’s risk aversion coefficient depends on the current wealth, the equilibrium strategy and the corresponding equilibrium value function are also derived. Our results show that with the same level of variance in the terminal wealth, the expected optimal terminal wealth under the precommitment strategy is greater than that under the equilibrium strategy with a constant risk aversion coefficient; the equilibrium strategy with a constant risk aversion coefficient is revealed to be different from that with a state-dependent risk aversion coefficient; and our results can also be degenerated to the results of He and Liang (2013b) and Björk et al. (2014). Finally, some numerical simulations are provided to illustrate our derived results.     

Optimal investment and risk control for an insurer under inside information

This paper is devoted to the study of the optimal investment and risk control strategy for an insurer who has some inside information on the financial market and the insurance business. The insurer’s risk process and the risky asset process in the financial market are assumed to be very general jump diffusion processes. The two processes are supposed to be correlated. Under the criterion of logarithmic utility maximization of the terminal wealth, we solve our problem by using forward integral approach. Some interesting particular cases are studied in which the explicit expressions of the optimal strategy are derived by using enlargement of filtration techniques.     

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.     

An adaptive algorithm for discrete-time differential games

This paper presents an algorithm for a player to improve his performance by adapting optimally over his non-optimally playing opponent in discrete-time differential games. The algorithm first estimates the opponent's actual strategies and then constructs an adaptive strategy for the player. The adaptive strategy is periodically updated according to the opponent's behavior using the neighboring optimal closed-loop solution technique. An example is given which demonstrates the superiority of this algorithm over the conventional one which assumes that the opponent plays optimally.     

低碳经济下三级供应链动态减排的微分博弈分析

在低碳经济的背景下,考虑到碳减排的连续性,引入时间因素,从动态角度将微分博弈的方法应用到三级供应链的减排决策中,构建了由一个供应商、一个制造商和一个零售商组成三级供应链的动态减排决策模型,比较分析在协同状态和非协同状态下产品减排量的变化曲线和供应链利润,同时还得到供应商、制造商和零售商的最优决策及利润。最后,使用算例分析的方法分析了各参数对产品减排量轨迹的影响,所得结论可以为供应商、制造商和零售商的决策提供理论依据。     

A Unified Approach for Modeling and Designing Attribute Sampling Plans for Monitoring Dependent Production Processes

In this paper, we consider a probabilistic model to represent some general dependent production processes and present a unified approach for designing attribute sampling plans for monitoring the ongoing production process. This model includes the classical iid model, independent model, Markov-dependent model and previous-sum dependent model, to mention a few. Some important properties of this model are established. We derive the recurrence relations for the probability distribution of the sum of n consecutive characteristics observed from the process. Using these recurrence relations, we present efficient algorithms for designing optimal single and double sampling plans for attributes, for monitoring the ongoing production process. Our algorithmic approach, which uses effectively the recurrence relations, yields a direct and an exact method, unlike many approximate methods adopted in the literature. Several interesting examples concerning specific models are discussed and a few tables for some special cases are also presented. It is demonstrated that the optimal double sampling plans lead to about 42% reduction in average sample number over the single sampling plans for process monitoring. AMS 2000 Subject Classifications: Primary 62P30; Secondary 62E15, 65C60     

One-armed bandit process with a covariate

We generalize the bandit process with a covariate introduced by Woodroofe in several significant directions: a linear regression model characterizing the unknown arm, an unknown variance for regression residuals and general discounting sequence for a non-stationary model. With the Bayesian regression approach, we assume a normal-gamma conjugate prior distribution of the unknown parameters. It is shown that the optimal strategy is determined by a sequence of index values which are monotonic and determined by the observed value of the covariate and updated posterior distributions. We further show that the myopic strategy is not optimal in general. Such structural properties help to understand the tradeoff between information gathering and immediate expected payoff and may provide certain insight for covariate adjusted response adaptive design of clinical trials.     

Solutions to Constrained Optimal Control Problems with Linear Systems

This paper is devoted to present solutions to constrained finite-horizon optimal control problems with linear systems, and the cost functional of the problem is in a general form. According to the Pontryagin’s maximum principle, the extremal control of such problem is a function of the costate trajectory, but an implicit function. We here develop the canonical backward differential flows method and then give the extremal control explicitly with the costate trajectory by canonical backward differential flows. Moreover, there exists an optimal control if and only if there exists a unique extremal control. We give the proof of the existence of the optimal solution for this optimal control problem with Green functions.     

Optimal dynamic asset allocation for DC plan accumulation/decumulation: Ambition-CVAR

We consider the late accumulation stage, followed by the full decumulation stage, of an investor in a defined contribution (DC) pension plan. The investor’s portfolio consists of a stock index and a bond index. As a measure of risk, we use conditional value at risk (CVAR) at the end of the decumulation stage. This is a measure of the risk of depleting the DC plan, which is primarily driven by sequence of return risk and asset allocation during the decumulation stage. As a measure of reward, we use Ambition, which we define to be the probability that the terminal wealth exceeds a specified level. We develop a method for computing the optimal dynamic asset allocation strategy which generates points on the efficient Ambition-CVAR frontier. By examining the Ambition-CVAR efficient frontier, we can determine points that are Median-CVAR optimal. We carry out numerical tests comparing the Median-CVAR optimal strategy to a benchmark constant proportion strategy. For a fixed median value (from the benchmark strategy) we find that the optimal Median-CVAR control significantly improves the CVAR. In addition, the median allocation to stocks at retirement is considerably smaller than the benchmark allocation to stocks.     

Robust optimal investment–reinsurance strategies for an insurer with multiple dependent risks

This paper considers a robust optimal investment and reinsurance problem with multiple dependent risks for an Ambiguity-Averse Insurer (AAI), who is uncertain about the model parameters. We assume that the surplus of the insurance company can be allocated to the financial market consisting of one risk-free asset and one risky asset whose price process satisfies square root factor process. Under the objective of maximizing the expected utility of the terminal surplus, by adopting the technique of stochastic control, closed-form expressions of the robust optimal strategy and the corresponding value function are derived. The verification theorem is also provided. Finally, by presenting some numerical examples, the impact of some parameters on the optimal strategy is illustrated and some economic explanations are also given. We find that the robust optimal reinsurance strategies under the generalized mean–variance premium are very different from that under the variance premium principle. In addition, ignoring model uncertainty risk will lead to significant utility loss for the AAI.     

Optimal control in a macroeconomic problem

The Pontryagin maximum principle is used to develop an original algorithm for finding an optimal control in a macroeconomic problem. Numerical results are presented for the optimal control and optimal trajectory of the development of a regional economic system. For an optimal control satisfying a certain constraint, an invariant of a macroeconomic system is derived.     

Time-consistent mean-variance reinsurance-investment in a jump-diffusion financial market

This paper study an optimal time-consistent reinsurance-investment strategy selection problem in a financial market with jump-diffusion risky asset, where the insurance risk model is modulated by a compound Poisson process. The aggregate claim process and the price process of risky asset are correlated by a common Poisson process. 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, we attack it by placing the problem within a game theoretic framework and looking for subgame perfect Nash equilibrium strategy. We investigate the problem using the extended Hamilton–Jacobi–Bellman dynamic programming approach. Closed-form solutions for the optimal reinsurance-investment strategy and the corresponding value functions are obtained. Numerical examples and economic significance analysis are also provided to illustrate how the optimal reinsurance-investment strategy changes when some model parameters vary.     

Optimal investment strategy for asset-liability management under the Heston model

ABSTRACT

This paper focuses on an asset-liability management problem for an investor who can invest in a risk-free asset and a risky asset whose price process is governed by the Heston model. The objective of the investor is to find an optimal investment strategy to maximize the expected exponential utility of the surplus process. By using the stochastic control method and variable change techniques, we obtain a closed-form solution of the corresponding Hamilton–Jacobi–Bellman equation. We also develop a verification theorem without the usual Lipschitz assumptions which can ensure that this closed-form solution is indeed the value function and then derive the optimal investment strategy explicitly. Finally, we provide numerical examples to show how the main parameters of the model affect the optimal investment strategy.     

Optimal control of a birth and death epidemic process

We employ a birth and death process to describe the spread of an infectious disease through a closed population. Control of the epidemic can be effected at any instant by varying the birth and death rates to represent quarantine and medical care programs. An optimal strategy is one which minimizes the expected discounted losses and costs resulting from the epidemic process and the control programs over an infinite horizon. We formulate the problem as a continuous-time Markov decision model. Then we present conditions ensuring that optimal quarantine and medical care program levels are nonincreasing functions of the number of infectives in the population. We also analyze the dependence of the optimal strategy on the model parameters. Finally, we present an application of the model to the control of a rumor.     

Optimal reinsurance-investment strategy for a dynamic contagion claim model

We study the optimal reinsurance-investment problem for the compound dynamic contagion process introduced by Dassios and Zhao (2011). This model allows for self-exciting and externally-exciting clustering effect for the claim arrivals, and includes the well-known Cox process with shot noise intensity and the Hawkes process as special cases. For tractability, we assume that the insurer’s risk preference is the time-consistent mean–variance criterion. By utilizing the dynamic programming and extended HJB equation approach, a closed-form expression is obtained for the equilibrium reinsurance-investment strategy. An excess-of-loss reinsurance type is shown to be optimal even in the presence of self-exciting and externally-exciting contagion claims, and the strategy depends on both the claim size and claim arrivals assumptions. Further, we show that the self-exciting effect is of a more dangerous nature than the externally-exciting effect as the former requires more risk management controls than the latter. In addition, we find that the reinsurance strategy does not always become more conservative (i.e., transferring more risk to the reinsurer) when the claim arrivals are contagious. Indeed, the insurer can be better off retaining more risk if the claim severity is relatively light-tailed.