多阶段均值–方差资产负债管理的随机控制

资产负债管理研究如何合理分配资产以到达最小化风险同时确保期望剩余财富(财富减去负债)达到一定水平.本文在均值–方差投资组合理论的框架下研究两类资产负债管理模型, 包括带有跨期均值–方差投资目标和带有非破产约束的模型. 由于在动态规划意义下, 方差不具有可分性质, 传统的随机最优控制方法难以直接应用. 如采用处理动态均值–方差优化问题的嵌入法来解决以上问题会带来计算上的困难. 本文借鉴平均场控制的思想对以上两类问题加以研究. 本文假设了非常宽泛的市场模型: 所有的资产都是风险资产; 债务和风险资产之间存在相关性. 在此市场假设模型下, 本文给出了最优投资策略(控制率)的解析表达式和均值–方差有效前沿的表达形式. 本研究成果为投资者提供了新的投资策略, 可应用于更复杂的资产负债管理中.     

基于随机基准的动态均值-方差投资组合选择

在不完全市场下, 研究基于随机基准的动态均值-方差投资组合选择问题. 该问题也可以理解为一个跟踪误差动态投资组合问题, 并将之转化为一个等价的考虑风险调整的期望相对收益最大化问题. 利用随机动态规划方法, 给出了最优投资策略和有效前沿的显式表达式. 最后通过实证分析表明了不完全市场和完全市场下最优投资策略和有效前沿的变化, 并对相关结论进行了经济解释.

     

不确定环境下的一种动态响应库存优化方法

针对随机需求提前期环境下的库存管理问题,提出一种启发式动态响应算法求取最优订货策略。建立最优订货策略的动态非线性优化模型,并设定客户需求和订货提前期分别为线性和高斯随机变量,通过变化形式、步长和变化频率的不同模拟实际经济运营过程;在微粒群寻优过程中引入柔性变异概率及动态更新响应方式,使微粒具有感知外界环境变化及对变化的响应能力,提高算法对复杂动态系统环境变化的适应性。实证分析结果证明了所提方法对最优订货量实时变化的动态响应能力。     

改进微分进化算法在软硬件划分中的应用

针对嵌入式系统的软硬件划分问题,在混合优化策略微分进化算法的基础上,提出一种采用随机变异策略的改进微分进化算法。在扩大最优向量影响范围的同时,减小变异向量受最优向量影响的程度。实验结果表明,与混合优化策略微分进化算法相比,该算法具有更快的收敛速度和更高的可靠性,能够较好地解决嵌入式系统的软硬件划分问题。     

多阶段均值-方差资产负债管理的随机控制（英文）

资产负债管理研究如何合理分配资产以到达最小化风险同时确保期望剩余财富(财富减去负债)达到一定水平.本文在均值-方差投资组合理论的框架下研究两类资产负债管理模型,包括带有跨期均值-方差投资目标和带有非破产约束的模型.由于在动态规划意义下,方差不具有可分性质,传统的随机最优控制方法难以直接应用.如采用处理动态均值-方差优化问题的嵌入法来解决以上问题会带来计算上的困难.本文借鉴平均场控制的思想对以上两类问题加以研究.本文假设了非常宽泛的市场模型:所有的资产都是风险资产;债务和风险资产之间存在相关性.在此市场假设模型下,本文给出了最优投资策略(控制率)的解析表达式和均值-方差有效前沿的表达形式.本研究成果为投资者提供了新的投资策略,可应用于更复杂的资产负债管理中.     

带Poisson跳的线性二次随机微分博弈及其在鲁棒控制中的应用

研究了一类带Poisson跳扩散过程的线性二次随机微分博弈,包括非零和博弈的Nash均衡策略与零和博弈的鞍点均衡策略问题．利用微分博弈的最大值原理,得到Nash均衡策略的存在条件等价于两个交叉耦合的矩阵Riccati方程存在解,鞍点均衡策略的存在条件等价于一个矩阵Riccati方程存在解的结论,并给出了均衡策略的显式表达及最优性能泛函值．最后,将所得结果应用于现代鲁棒控制中的随机H₂/H_∞控制与随机H_∞控制问题,得到了鲁棒控制策略的存在条件及显式表达,并验证所得结果在金融市场投资组合优化问题中的应用．     

求解非光滑强凸优化问题的减小方差加权随机算法*

在光滑问题随机方法中使用减小方差策略,能够有效改善算法的收敛效果.文中同时引用加权平均和减小方差的思想,求解“L1+L2+Hinge”非光滑强凸优化问题,得到减小方差加权随机算法(α-HRMDVR-W).在每步迭代过程中使用减小方差策略,并且以加权平均的方式输出,证明其具有最优收敛速率,并且该收敛速率不依赖样本数目.与已有减小方差方法相比,α-HRMDVR-W每次迭代中只使用部分样本代替全部样本修正梯度.实验表明α-HRMDVR-W在减小方差的同时也节省CPU时间.     

机构投资者的最优变现策略

相对于普通投资者,大规模持有某种资产的机构投资者在交易这种资产时,其行为会导致资产价格的单向变动,从而产生流动性风险。针对机构投资者在连续时间框架且股票价格服从几何布朗运动的情况,提出在随机冲击下其完全变现行为的最优变现策略,利用最优控制理论中的极小值原理研究其最优变现策略。敏感性分析表明,最优变现策略由市场价格波动率、资产的流动性和机构投资者的风险厌恶偏好共同决定。以深发展股票为例,验证了机构投资者在随机冲击下可根据上述结论选择合适的最优变现策略,降低其在市场中所面临的流动性风险,使得股票的账面价值更可能多的转换为实际收益。     

随机金融市场环境下的最优再保险–投资策略

为了对冲保险风险,保险公司可以向再保险公司购买比例再保险;同时,为了保值增值,保险公司将其财富投资于金融市场.假设盈余过程由带漂移的布朗运动所驱动,利率满足仿射利率模型,股票波动率满足Heston随机波动率模型.应用随机最优控制和HJB方程方法得到了指数效用下最优再保险–投资策略的显式解.给出数值算例并分析了模型参数对最优再保险策略和最优投资策略的影响.研究结果表明:最优再保险策略不仅依赖于保险市场参数,而且依赖于金融市场参数;随机利率与随机波动率模型下的最优再保险–投资策略与利率动态密切相关,而与波动率动态无关;再保险行为对投资于股票的数量没有影响,而对投资于零息票债券的数量产生较大的影响.     

马科维茨均值-方差理论在能源期货投资组合优化中的运用

本文以个人投资者的视角,探索能源期货投资策略。以马科维茨均值-方差理论为依据,以4种能源期货为标的资产,运用MATLAB软件进行计算,构造了区间为第n日至n+252日的动态最佳投资比率策略,以第n+1日至n+253日作为检验区间,并对构造区间、检验区间滚动k次,验证了该投资比率策略的有效性及动态策略的平稳性。投资策略的有效性表明,它是投资者进行能源期货投资的一种可行思路。同时,也验证了马科维茨均值-方差理论可运用于数量较少的能源期货投资组合中。     

不确定拟哈密顿系统的随机最优控制

本文提出了不确定拟哈密顿系统、基于随机平均法、随机极大值原理和随机微分对策理论的一种随机极大极小最优控制策略.首先,运用拟哈密顿系统的随机平均法,将系统状态从速度和位移的快变量形式转化为能量的慢变量形式,得到部分平均的It随机微分方程;其次,给定控制性能指标,对于不确定拟哈密顿系统的随机最优控制,根据随机微分对策理论,将其转化为一个极小极大控制问题;再根据随机极大值原理,建立关于系统与伴随过程的前向-后向随机微分方程,随机最优控制表达为哈密顿控制函数的极大极小条件,由此得到最坏情形下的扰动参数与极大极小最优控制;然后,将最坏扰动参数与最优控制代入部分平均的It随机微分方程并完成平均,求解与完全平均的It随机微分方程相应的Fokker-Planck-Kolmogorov(FPK)方程,可得受控系统的响应量并计算控制效果;最后,将上述不确定拟哈密顿系统的随机最优控制策略应用于一个两自由度非线性系统,通过数值结果说明该随机极大极小控制策略的控制效果.     

Feasible solution to discrete-time linear quadratic stochastic Stackelberg difference game

This paper investigates the discrete-time linear quadratic (LQ) stochastic Stackelberg game, which has not been thoroughly addressed in previous literature. Firstly, we derive the maximum principle for the stochastic Stackelberg difference game using the variational method, and obtain the necessary and sufficient solvability conditions. However, due to the coupling between the two players and the presence of stochastic noise, obtaining explicit optimal leader and follower's strategies becomes challenging. Therefore, we present a feasible suboptimal control strategy instead. As a result, we derive a feasible suboptimal control strategy. To achieve this, we assume a linear homogeneous relationship to decouple the group of stochastic game forward-backward stochastic differential equations (SG-FBSDEs), which serves as a compromise for obtaining the optimal solution. With this approach, we derive a feasible solution to the stochastic Stackelberg difference game based on the solution to symmetric Riccati equations.     

A game-theoretic method for cross-layer stochastic resilient control design in CPS

In this paper, the cross-layer security problem of cyber-physical system (CPS) is investigated from the game-theoretic perspective. Physical dynamics of plant is captured by stochastic differential game with cyber-physical influence being considered. The sufficient and necessary condition for the existence of state-feedback equilibrium strategies is given. The attack-defence cyber interactions are formulated by a Stackelberg game intertwined with stochastic differential game in physical layer. The condition such that the Stackelberg equilibrium being unique and the corresponding analytical solutions are both provided. An algorithm is proposed for obtaining hierarchical security strategy by solving coupled games, which ensures the operational normalcy and cyber security of CPS subject to uncertain disturbance and unexpected cyberattacks. Simulation results are given to show the effectiveness and performance of the proposed algorithm.     

Dynamic option hedging with transaction costs: A stochastic model predictive control approach

This paper proposes stochastic model predictive control as a tool for hedging derivative contracts (such as plain vanilla and exotic options) in the presence of transaction costs. The methodology combines stochastic scenario generation for the prediction of asset prices at the next rebalancing interval with the minimization of a stochastic measure of the predicted hedging error. We consider 3 different measures to minimize in order to optimally rebalance the replicating portfolio: a trade‐off between variance and expected value of hedging error, conditional value at risk, and the largest predicted hedging error. The resulting optimization problems require solving at each trading instant a quadratic program, a linear program, and a (smaller‐scale) linear program, respectively. These can be combined with 3 different scenario generation schemes: the lognormal stock model with parameters recursively identified from data, an identification method based on support vector regression, and a simpler scheme based on perturbation noise. The hedging performance obtained by the proposed stochastic model predictive control strategies is illustrated on real‐world data drawn from the NASDAQ‐100 composite, evaluated for a European call and a barrier option, and compared with delta hedging.     

Necessary and sufficient conditions of risk‐sensitive optimal control and differential games for stochastic differential delayed equations

In this paper, we consider risk‐sensitive optimal control and differential games for stochastic differential delayed equations driven by Brownian motion. The problems are related to robust stochastic optimization with delay due to the inherent feature of the risk‐sensitive objective functional. For both problems, by using the logarithmic transformation of the associated risk‐neutral problem, the necessary and sufficient conditions for the risk‐sensitive maximum principle are obtained. We show that these conditions are characterized in terms of the variational inequality and the coupled anticipated backward stochastic differential equations (ABSDEs). The coupled ABSDEs consist of the first‐order adjoint equation and an additional scalar ABSDE, where the latter is induced due to the nonsmooth nonlinear transformation of the adjoint process of the associated risk‐neutral problem. For applications, we consider the risk‐sensitive linear‐quadratic control and game problems with delay, and the optimal consumption and production game, for which we obtain explicit optimal solutions.     

LSV models with stochastic interest rates and correlated jumps

Pricing and hedging exotic options using local stochastic volatility models drew a serious attention within the last decade, and nowadays became almost a standard approach to this problem. In this paper we show how this framework could be extended by adding to the model stochastic interest rates and correlated jumps in all three components. We also propose a new fully implicit modification of the popular Hundsdorfer and Verwer and Modified Craig–Sneyd finite-difference schemes which provides second-order approximation in space and time, is unconditionally stable and preserves positivity of the solution, while still has a linear complexity in the number of grid nodes.     

网格资源分配博弈的随机动态分析*

针对网格资源分配的优化问题,提出利用随机动态来研究有限网格群体博弈的分析方法。通过建立网格使用者策略选择的随机模型来分析有限网格群体的博弈,并利用期望效用生成选择过程的量化指标来判断使用者在反复博弈中策略选择的变化方向及其稳定性。最后通过仿真实例的研究结果表明,在效用矩阵不变的情况下,群体规模是影响网格使用者策略选择方案的一个重要因素。     

Stochastic differential games and inverse optimal control and stopper policies

In this paper, we consider a two-player stochastic differential game problem over an infinite time horizon where the players invoke controller and stopper strategies on a nonlinear stochastic differential game problem driven by Brownian motion. The optimal strategies for the two players are given explicitly by exploiting connections between stochastic Lyapunov stability theory and stochastic Hamilton–Jacobi–Isaacs theory. In particular, we show that asymptotic stability in probability of the differential game problem is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution to the steady-state form of the stochastic Hamilton–Jacobi–Isaacs equation, and hence, guaranteeing both stochastic stability and optimality of the closed-loop control and stopper policies. In addition, we develop optimal feedback controller and stopper policies for affine nonlinear systems using an inverse optimality framework tailored to the stochastic differential game problem. These results are then used to provide extensions of the linear feedback controller and stopper policies obtained in the literature to nonlinear feedback controllers and stoppers that minimise and maximise general polynomial and multilinear performance criteria.     

NONLINEAR DISCRETE-TIME RISK-SENSITIVE OPTIMAL CONTROL

This paper is devoted to the study of the connections among risk-sensitive stochastic optimal control, dynamic game optimal control, risk-neutral stochastic optimal control and deterministic optimal control in a nonlinear, discrete-t ime context with complete state information. The analysis worked out sheds light on the profound links among these control strategies, which remain hidden in the linear context. In particular, it is shown that, under suitable parameterizations, risk-sensi tive control can be regarded as a control methodology which combines features of both stochastic risk-neutral control and deterministic dynamic game control.