Maximum Principle for Markov Regime-Switching Forward–Backward Stochastic Control System with Jumps and Relation to Dynamic Programming

This paper presents a sufficient stochastic maximum principle for a stochastic optimal control problem of Markov regime-switching forward–backward stochastic differential equations with jumps. The relationship between the stochastic maximum principle and the dynamic programming principle in a Markovian case is also established. Finally, applications of the main results to a recursive utility portfolio optimization problem in a financial market are discussed.     

Nonsmooth maximum principle for infinite-horizon problems

In this paper, we consider a class of infinite-horizon discounted optimal control problems with nonsmooth problem data. A maximum principle in terms of differential inclusions with a Michel type transversality condition is given. It is shown that, when the discount rate is sufficiently large, the problem admits normal multipliers and a strong transversality condition holds. A relationship between dynamic programming and the maximum principle is also given.The author is indebted to Francis Clarke for helpful suggestions and discussions.     

Relationship Between MP and DPP for the Stochastic Optimal Control Problem of Jump Diffusions

This paper is concerned with the stochastic optimal control problem of jump diffusions. The relationship between stochastic maximum principle and dynamic programming principle is discussed. Without involving any derivatives of the value function, relations among the adjoint processes, the generalized Hamiltonian and the value function are investigated by employing the notions of semijets evoked in defining the viscosity solutions. Stochastic verification theorem is also given to verify whether a given admissible control is optimal.     

VISCOSITY SOLUTIONS OF HJB EQUATIONS ARISING FROM THE VALUATION OF EUROPEAN PASSPORT OPTIONS

The passport option is a call option on the balance of a trading account. The option holder retains the gain from trading, while the issuer is liable for the net loss. In this article, the mathematical foundation for pricing the European passport option is established. The pricing equation which is a fully nonlinear equation is derived using the dynamic programming principle. The comparison principle, uniqueness and convexity preserving of the viscosity solutions of related H J13 equation are proved. A relationship between the passport and lookback options is discussed.     

A new look at Bellman's principle of optimality

New light is shed on Bellman's principle of optimality and the role it plays in Bellman's conception of dynamic programming. It is argued that a failure to recognize the special features of the model in the context of which the principle was stated has resulted in the latter being misconstrued in the dynamic programming literature.     

Value Functions and Transversality Conditions for Infinite-Horizon Optimal Control Problems

This paper investigates a relationship between the maximum principle with an infinite horizon and dynamic programming and sheds new light upon the role of the transversality condition at infinity as necessary and sufficient conditions for optimality with or without convexity assumptions. We first derive the nonsmooth maximum principle and the adjoint inclusion for the value function as necessary conditions for optimality. We then present sufficiency theorems that are consistent with the strengthened maximum principle, employing the adjoint inequalities for the Hamiltonian and the value function. Synthesizing these results, necessary and sufficient conditions for optimality are provided for the convex case. In particular, the role of the transversality conditions at infinity is clarified.     

Multitime Dynamic Programming for Curvilinear Integral Actions

This paper justifies dynamic programming PDEs for optimal control problems with performance criteria involving curvilinear integrals. The main novel feature, relative to the known theory, is that the multitime dynamic programming PDEs are now connected to the multitime maximum principle. For the first time, an interesting and useful connection between the multitime maximum principle and the multitime dynamic programming is given, characterizing the optimal control by means of a PDE system that may be viewed as a multitime feedback law.     

Multitime dynamic programming for multiple integral actions

This paper introduces a new type of dynamic programming PDE for optimal control problems with performance criteria involving multiple integrals. The main novel feature of the multitime dynamic programming PDE, relative to the standard Hamilton-Jacobi-Bellman PDE, is that it is connected to the multitime maximum principle and is of divergence type. Introducing a generating vector field for the maximum value function, we present an interesting and useful connection between the multitime maximum principle and the multitime dynamic programming, characterizing the optimal control by means of a multitime Hamilton-Jacobi-Bellman (divergence) PDE that may be viewed as a feedback law. Section 1 recalls the multitime maximum principle. Section 2 shows how a multitime control dynamics determines the multitime Hamilton-Jacobi-Bellman PDE via a generating vector field of the value function. Section 3 gives an example of two-time dynamics with nine velocities proving that our theory works well. Section 4 shows that the Hamilton PDEs are characteristic PDEs of multitime Hamilton-Jacobi PDE and that the costates in the multitime maximum principle are in fact gradients of the components of the generating vector field.     

多目标规划的极大熵方法

多目标规划的极大熵方法王雪华，秦学志（大连理工大学）ＴＨＥＭＡＸＩＭＵＭＥＮＴＲＯＰＹＭＥＴＨＯＤＦＯＲＭＵＬＴＩＯＢＪＥＣＴＩＶＥＰＲＯＧＲＡＭＭＩＮＧ￥ＷａｎｇＸｕｅ－ｈｕａ；ＱｉｎＸｕｅ－ｚｈｉ（ＤａｌｉａｎＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏ...     

Linear Optimal Control Problems and Generalized Linear Programs

Linear optimal control problems with state inequality constraints is an important class of large systems. This paper shows that a generalized programming formulation of these problems does not result in a decomposition over time or a maximum principle as it does for problems without the state constraints. A semi-finite generalized programming formulation, however, can be used to formally produce the maximum principle, i.e. the necessary optimality conditions, for problems under consideration.     

Stable sequential convex programming in a Hilbert space and its application for solving unstable problems

A parametric convex programming problem with an operator equality constraint and a finite set of functional inequality constraints is considered in a Hilbert space. The instability of this problem and, as a consequence, the instability of the classical Lagrange principle for it is closely related to its regularity and the subdifferentiability properties of the value function in the optimization problem. A sequential Lagrange principle in nondifferential form is proved for the indicated convex programming problem. The principle is stable with respect to errors in the initial data and covers the normal, regular, and abnormal cases of the problem and the case where the classical Lagrange principle does not hold. It is shown that the classical Lagrange principle in this problem can be naturally treated as a limiting variant of its stable sequential counterpart. The possibility of using the stable sequential Lagrange principle for directly solving unstable optimal control problems and inverse problems is discussed. For two illustrative problems of these kinds, the corresponding stable Lagrange principles are formulated in sequential form.     

一种用于线性规划算法的换基准则

本文就线性规划中的对偶单纯形法和运输问题中的表上作业法选取出基变量或者对基变量的准则进行改进,从而得出一种新的换基准则.按该方法进行优化运算,可以使算法的迭代次数减到最少,从而加快了运算速度.     

Linear programming approaches for multicategory support vector machines

We consider linear programming approaches for support vector machines (SVM). The linear programming problems are introduced as an approximation of the quadratic programming problems commonly used in SVM. When we consider the kernel based nonlinear discriminators, the approximation can be viewed as kernel principle component analysis which generates an important subspace from the feature space characterized the kernel function. We show that any data points nonlinearly, and implicitly, projected into the feature space by kernel functions can be approximately expressed as points lying a low dimensional Euclidean space explicitly, which enables us to develop linear programming formulations for nonlinear discriminators. We also introduce linear programming formulations for multicategory classification problems. We show that the same maximal margin principle exploited in SVM can be involved into the linear programming formulations. Moreover, considering the low dimensional feature subspace extraction, we can generate nonlinear multicategory discriminators by solving linear programming problems.Numerical experiments on real world datasets are presented. We show that the fairly low dimensional feature subspace can achieve a reasonable accuracy, and that the linear programming formulations calculate discriminators efficiently. We also discuss a sampling strategy which might be crucial for huge datasets.     

基于结构元方法的可能性线性规划

主要目的是利用结构元方法来解决含有模糊系数的线性规划问题,即可能性线性规划问题.首先,简单地介绍了结构元方法及结构元加权序,证明了其模糊优先的合理性,并同原有序关系进行了比较.然后,利用这种序关系,将可能性线性规划问题等价地转化为一个经典的线性规划问题,简化了原问题的求解.最后,借助一个实际例子,进一步表明了该方法的有效性.     

线性规划问题的一种转换解法

本文根据矩阵对策与对偶规划问题的等价性，以及矩阵对策的凌越原则，对线性规划问题提出了一种转换解法。     

非线性不等式与等式约束的多目标规划问题的区间极大熵方法

利用极大熵方法将带多个非线性不等式约束和多个非线性等式约束的多目标规划问题变为两个非线性不等式约束的单个可微的目标函数优化问题,并结合区间分析知识给出一种新的解决多目标规划问题的区间方法.     

Stochastic Maximum Principle for Forward-Backward Regime Switching Jump Diffusion Systems and Applications to Finance

The authors prove a sufficient stochastic maximum principle for the optimal control of a forward-backward Markov regime switching jump diffusion system and show its connection to dynamic programming principle. The result is applied to a cash flow valuation problem with terminal wealth constraint in a financial market. An explicit optimal strategy is obtained in this example.     

求解不确定型二层线性规划模型的悲观决策方法

基于多参数线性规划理论，将不确定型二层线性规划问题转化为多个关于不确定参数的线性规划问题。利用不确定型决策方法中的悲观准则．从最不利的结果中选择最有利的结果，从而得到不确定型二层线性规划的最优解。数值实例的仿真结果表明，所提出的悲观决策方法对解决诸如不确定供应链的规划与运作等问题不失为一种有效的决策支持工具。     

Maximum principle,dynamic programming,and their connection in deterministic control

Two major tools for studying optimally controlled systems are Pontryagin's maximum principle and Bellman's dynamic programming, which involve the adjoint function, the Hamiltonian function, and the value function. The relationships among these functions are investigated in this work, in the case of deterministic, finite-dimensional systems, by employing the notions of superdifferential and subdifferential introduced by Crandall and Lions. Our results are essentially non-smooth versions of the classical ones. The connection between the maximum principle and the Hamilton-Jacobi-Bellman equation (in the viscosity sense) is thereby explained by virtue of the above relationship.This research was supported by the Natural Science Fund of China.This paper was written while the author visited Keio University, Japan. The author is indebted to Professors H. Tanaka and M. Nisio for their helpful suggestions and discussions. Thanks are also due to Professor X. J. Li for his comments and criticism.     

Robust Optimal Investment Problem with Delay under Heston’s Model

This paper considers a robust optimal portfolio problem under Heston model in which the risky asset price is related to the historical performance. The finance market includes a riskless asset and a risky asset whose price is controlled by a stochastic delay equation. The objective is to choose the investment strategy to maximize the minimal expected utility of terminal wealth. By employing dynamic programming principle and Hamilton-Jacobin-Bellman (HJB) equation, we obtain the specific expression of the optimal control and the explicit solution of the corresponding HJB equation. Besides, a verification theorem is provided to ensure the value function is indeed the solution of the HJB equation. Finally, we use numerical examples to illustrate the relationship between the optimal strategy and parameters.