具有Markov跳跃参数的一类随机非线性系统逆最优增益设计

研究了一类随机非线性系统的逆最优增益设计问题,系统中除了方差未知的Wiener噪声之外,还含有Markov跳跃参数. 首先,给出此类系统逆最优增益设计问题可解的一个充分条件. 其次,针对一类具有严格反馈形式的随机非线性系统,利用积分反推法,给出了依概率全局渐近稳定和逆最优控制策略的设计方法. 其中,所设计的Lyapunov函数和控制策略与模态显式无关,克服了由于Markov跳跃模态引起的耦合项所带来的设计困难. 最后,通过仿真验证了控制策略的有效性.     

考虑通货膨胀因素下的连续时间均值-方差投资组合选择

考虑通货膨胀因素,利用均值-方差模型研究连续时间投资组合选择问题.利用 Lagrange 乘子技术将原均值-方差模型转化为一个标准的随机最优控制问题,应用动态规划的方法得到问题的解析解,进而求解出原均值-方差模型的有效投资策略和有效边界的解析表达式.通过实证分析进一步表明了结论的正确性.     

一类带随机跳跃的完全耦合的线性二次随机控制问题

研究一类带随机跳跃的完全耦合的线性二次随机控制问题. 得到了最优控制的显式解, 并可以证明最优控制是唯一的. 引入了一类推广的黎卡提方程并讨论了其可解性. 利用这一类推广的黎卡提方程的解, 得到了上述带随机跳跃的最优控制问题的线性状态反馈调节器.     

带有随机跳跃干扰的线性二次随机最优控制问题

给出一类布朗运动和泊松过程混合驱动的正倒向随机微分方程解的存在唯一性结果, 应用这一结果研究带有随机跳跃干扰的线性二次随机最优控制问题,并得到最优控制的显式形 式,可以证明最优控制是唯一的.然后,引入和研究一类推广的黎卡提方程系统,讨论该方程系统 的可解性并由该方程的解得到带有随机跳跃干扰的线性二次随机最优控制问题最优的线性反馈.     

疫情沿交通工具跨区域扩散建模及最优控制

交通运输系统在服务民众的同时也为疫情沿跨区域传播扩散提供了载体.鉴于此,研究疫情沿多种交通工具跨区域扩散模型及最优控制问题.考虑人口空间状态、迁徙过程、疫情状态和交通工具特征,构建多区域迁徙-疫情扩散耦合动力学方程,并分析其扩散性质.综合考虑应急资源的有限性,进一步建立基于本地与迁徙隔离政策的动态最优控制模型.数值计算对比分析不同管控策略组合下疫情的扩散速度与范围,验证疫情扩散模型和最优控制策略的有效性.结果表明:疫情可以借助交通工具快速扩散;仅对单一交通工具实施管控措施可以快速降低疫情跨区域扩散的速度,但对均衡状态的扩散范围影响小.     

离散非线性零和博弈的事件驱动最优控制方案

在求解离散非线性零和博弈问题时,为了在有效降低网络通讯和控制器执行次数的同时保证良好的控制效果,本文提出了一种基于事件驱动机制的最优控制方案.首先,设计了一个采用新型事件驱动阈值的事件驱动条件,并根据贝尔曼最优性原理获得了最优控制对的表达式.为了求解该表达式中的最优值函数,提出了一种单网络值迭代算法.利用一个神经网络构建评价网.设计了新的评价网权值更新规则.通过在评价网、控制策略及扰动策略之间不断迭代,最终获得零和博弈问题的最优值函数和最优控制对.然后,利用Lyapunov稳定性理论证明了闭环系统的稳定性.最后,将该事件驱动最优控制方案应用到了两个仿真例子中,验证了所提方法的有效性.     

一类部分信息下带有劳动力收入的最优投资消费问题

最优投资消费问题属于一类典型的随机最优控制问题. 劳动力收入可通过影响期望效用从而影响投资消 费策略的制定. 本文首次在股票收益率和劳动力收入均为不可观测过程情形下, 研究了一类部分信息下的最优投资 消费问题. 首先综合运用Kalman滤波和非线性滤波, 得到了Zakai方程的显式解, 将部分信息下的随机最优控制问题 转化为完备信息下的随机最优控制问题. 其次通过求解HJB方程以及证明验证定理, 得到了该类最优投资消费问题 的最优策略以及值函数的显式表达. 最后采用真实市场数据进行仿真, 对比经典完备信息模型与本文部分信息模型 所得最优策略的差异, 验证了本文所得最优策略在有效利用市场信息方面的优越性.     

年龄相关的种群空间扩散系统的广义解与收获控制

研究了由积分偏微分方程描述的年龄相关的种群空间扩散系统的收获控制问题.首先利用不动点方法证明了对于有界死亡率μ的系统广义解的存在性,但这是预备的结果.进一步,运用上述结果、先验估计和紧性定理,证明了对于在r=A附近无界的μ的系统解的存在惟一性.其次,利用类似方法得到系统最优收获控制的存在性.最后,利用G^ateax微分和Lions的变分不等式理论,推得了控制为最优的必要条件;从而得到了由积分偏微分方程和变分不等式构成的最优性组.最优性组能够确定最优控制.还建立了表征最优控制的Euler_Lagrange组.这些结果可为种群系统控制问题的实际研究作为理论参考.     

含扩散项不可靠生产系统最优生产控制的数值求解

针对含扩散项不可靠随机生产系统最优生产控制的优化命题, 采用数值解方法来求解该优化命题最优控制所满足的模态耦合的非线性偏微分HJB方程. 首先构造Markov链来近似生产系统状态演化, 并基于局部一致性原理, 把求解连续时间随机控制问题转化为求解离散时间的Markov决策过程问题, 然后采用数值迭代和策略迭代算法来实现最优控制数值求解过程. 文末仿真结果验证了该方法的正确性和有效性.     

基于Radau伪谱法的非线性最优控制问题的收敛性

在过去的10年里,伪谱方法(如Legendre伪谱法、Gauss伪谱法、Radau伪谱法)逐步成为求解不同领域中非线性最优控制问题的一种高效、灵活的数值解法.本文从最优控制问题解的存在性、收敛性以及解的可行性3个方面对采用Radau伪谱法求解一般非线性最优控制问题解的收敛性进行研究.证明了原最优控制问题的离散解存在、存在收敛到原最优控制问题解上的离散解和离散形式的收敛解是原最优控制问题的最优解.在此基础上,证明了Radau伪谱法的收敛性.本文结论与现有文献相比,去掉了一些必要条件,更适合一般的非线性时不变系统.     

Modelling on optimal portfolio with exchange rate based on discontinuous stochastic process

Considering the stochastic exchange rate, this paper is concerned with the dynamic portfolio selection in financial market. The optimal investment problem is formulated as a continuous-time mathematical model under mean-variance criterion. These processes follow jump-diffusion processes (Weiner process and Poisson process). Then the corresponding Hamilton–Jacobi–Bellman(HJB) equation of the problem is presented and its efferent frontier is obtained. Moreover, the optimal strategy is also derived under safety-first criterion.     

Numerical solution of continuous-time mean–variance portfolio selection with nonlinear constraints

An investment problem is considered with dynamic mean–variance (M–V) portfolio criterion under discontinuous prices described by jump-diffusion processes. Some investment strategies are restricted in the study. This M–V portfolio with restrictions can lead to a stochastic optimal control model. The corresponding stochastic Hamilton–Jacobi–Bellman equation of the problem with linear and nonlinear constraints is derived. Numerical algorithms are presented for finding the optimal solution in this article. Finally, a computational experiment is to illustrate the proposed methods by comparing with M–V portfolio problem which does not have any constraints.     

A class of continuous-time portfolio selection with liability under jump-diffusion processes

A continuous-time mean-variance portfolio selection model is formulated with multiple risky assets and one liability under discontinuous prices which follow jump-diffusion processes in an incomplete market. The correlations between the risky assets and the liability are considered. The corresponding Hamilton–Jacobi–Bellman equation of the problem is presented. The optimal dynamic strategy and the efficient frontier in closed forms are derived explicitly by using stochastic linear-quadratic control technique. Finally, the effects on efficient frontier under the value-at-risk constraint are illustrated.     

Verification Theorem Of Stochastic Optimal Control With Mixed Delay And Applications To Finance

This paper focuses on a general model of a controlled stochastic differential equation with mixed delay in the state variable. Based on the Itô formula, stochastic analysis, convex analysis, and inequality technique, we obtain a semi‐coupled forward‐backward stochastic differential equation with mixed delay and mixed initial‐terminal conditions and prove that such forward‐backward system admits a unique adapted solution. The verification theorem for an optimal control of a system with mixed delay is established. The obtained results generalize and improve some recent results, and they are more easily verified and applied in practice. As an application, we conclude with finding explicitly the optimal consumption rate from the wealth process of a person given by a stochastic differential equation with mixed delay which fit into our general model.     

Partially observed non-linear risk-sensitive optimal stopping control for non-linear discrete-time systems

In this paper we introduce and solve the partially observed optimal stopping non-linear risk-sensitive stochastic control problem for discrete-time non-linear systems. The presented results are closely related to previous results for finite horizon partially observed risk-sensitive stochastic control problem. An information state approach is used and a new (three-way) separation principle established that leads to a forward dynamic programming equation and a backward dynamic programming inequality equation (both infinite dimensional). A verification theorem is given that establishes the optimal control and optimal stopping time. The risk-neutral optimal stopping stochastic control problem is also discussed.     

A Computational Method for Stochastic Optimal Control Problems in Financial Mathematics

Principle of optimality or dynamic programming leads to derivation of a partial differential equation (PDE) for solving optimal control problems, namely the Hamilton‐Jacobi‐Bellman (HJB) equation. In general, this equation cannot be solved analytically; thus many computing strategies have been developed for optimal control problems. Many problems in financial mathematics involve the solution of stochastic optimal control (SOC) problems. In this work, the variational iteration method (VIM) is applied for solving SOC problems. In fact, solutions for the value function and the corresponding optimal strategies are obtained numerically. We solve a stochastic linear regulator problem to investigate the applicability and simplicity of the presented method and prove its convergence. In particular, for Merton's portfolio selection model as a problem of portfolio optimization, the proposed numerical method is applied for the first time and its usefulness is demonstrated. For the nonlinear case, we investigate its convergence using Banach's fixed point theorem. The numerical results confirm the simplicity and efficiency of our method.     

Maximum principle for mean-field jump–diffusion stochastic delay differential equations and its application to finance

This paper investigates a stochastic optimal control problem with delay and of mean-field type, where the controlled state process is governed by a mean-field jump–diffusion stochastic delay differential equation. Two sufficient maximum principles and one necessary maximum principle are established for the underlying system. As an application, a bicriteria mean–variance portfolio selection problem with delay is studied to demonstrate the effectiveness and potential of the proposed techniques. Under certain conditions, explicit expressions are provided for the efficient portfolio and the efficient frontier, which are as elegant as those in the classical mean–variance problem without delays.     

UNCERTAIN OPTIMAL CONTROL WITH APPLICATION TO A PORTFOLIO SELECTION MODEL

Optimal control is a very important field of study not only in theory but in applications, and stochastic optimal control is also a significant branch of research in theory and applications. Based on the concept of uncertain process, an uncertain optimal control problem is dealt with. Applying Bellman's principle of optimality, the principle of optimality for uncertain optimal control is obtained, and then a fundamental result called the equation of optimality in uncertain optimal control is given. Finally, as an application, the equation of optimality is used to solve a portfolio selection model.     

Optimal Control Problem for Risk‐Sensitive Mean‐Field Stochastic Delay Differential Equation with Partial Information

This paper deals with the risk‐sensitive control problem for mean‐field stochastic delay differential equations (MF‐SDDEs) with partial information. Firstly, under the assumptions that the control domain is not convex and the value function is non‐smooth, we establish a stochastic maximum principle (SMP). Then, by means of Itô's formula and some continuous dependence, we prove the existence and uniqueness results for another type of MF‐SDDEs. Meanwhile, the verification theorem for the MF‐SDDEs is obtained by using a clever construction of the Hamiltonian function. Finally, based on our verification theorem, a linear‐quadratic system is investigated and the optimal control is also derived by the stochastic filtering technique.