带随机过程的随机规划问题最优解集的过程特性与稳定性

本文证明了带随机过程的随机规划问题最优解集做为集值随机过程的可测性、可测最优解选择过程的存在性。研究了最优解集过程的平稳性、马氏性以及最优值过程的鞅性和最优解集过程的集值鞅性。最后，讨论了在有限维分布意义下最优解集过程对所含随机过程参数的连续性以及最优值过程的稳定性。     

带随机过程的随机规划问题——最优值过程与最优解集过程的鞅性

假设问题中所含随机过程为鞅，本文证明了带随机过程的随机规划问题共最优值过程与最优解集过程分别为实值上鞅与集值上鞅，且存在最优鞅通过程。     

一类随机过程的经验测度的大偏差定理

本文证明了一类随机过程的经验测度的大偏差定理，其中这类随机过程是一个随机发展过程解的小扰动．     

参变随机过程与重随机参变过程的若干应用

参变随机过程(见[4]、[6],简记为 PRP)是通常随机过程及多指标过程概念的推广,而重随机参变过程(见[5],简记为 DRPP)则包含随机环境中的随机过程(RWIRE,见[1]、[2])以及随机对策为其特例,PRP 与 DRPP 理论是针对广泛的实际背景提出的,虽然还不成熟,但已可找到若干应用.本文将对此作概略的初步探讨.§1先对 PRP、DRPP 以及估量概率的概念作简单介绍;§2—§6分别概述 PRP 与 DRPP 在竞赛模型与随机对策、体育竞赛的现场指导、仿型预测与控制、选材模型等优化问题方面的应用。§7简述 PRP 与 DRPP 观点下的 RWIRE、Bayes 估计以及气象预报的某些方法.     

两指标随机过程的最优停止的构造

本文研究了离散两指标随机过程X=(X_z,_z,z∈N~2)的最优停止的结构及X的Snell包络的渐近算法。首先证明了在条件(A~+)下:Γ=(γ_z,_z,z∈N~2)是控制X的最小正则上鞅,这里γ_z=E(X_σ|_z),z∈N~2。然后根据最优原理,利用X的Snell包络构造出最优策略。从而得出了报酬过程为X=(X_z,_z,z∈N~2)的最优停点的具体结构。最后证明了X的Snell包络的三重极限定理。     

带Lévy过程的正倒向对偶系统随机控制问题

讨论了一类控制系统是带Lévy过程的正倒向对偶随机微分方程的随机控制问题.本文假定控制区域为凸集,最优解是使目标函数达到最小的控制过程.使用带Lévy过程的Ito公式及Ekeland变分原理,作者建立了这类随机控制问题极值原理的一个必要条件.     

二层随机规划逼近最优解集的上半收敛性

下层随机规划以上层决策变量作为参数,而上层随机规划是以下层随机规划的唯一最优解作为响应的一类二层随机规划问题,首先在下层随机规划的原问题有唯一最优解的假设下,讨论了下层随机规划的任意一个逼近最优解序列都收敛于原问题的唯一最优解,然后将下层随机规划的唯一最优解反馈到上层,得到了上层随机规划逼近最优解集序列的上半收敛性.     

一般状态空间马氏过程随机泛函的矩

研究了一般状态空间马氏过程随机泛函的矩,利用最小非负解理论,得到了随机泛函的矩是相应方程的最小非负解.     

关于带跳的反射扩散过程的随机最优控制问题

研究一类半空间上带泊松跳的反射扩散过程的随机最优控制问题·　得到关于这一控制问题的非线性Nisio半群 ,和联系这一半群的带Neumann边界条件的哈密顿·雅可比·贝尔曼方程·　讨论这一类方程的粘性解的存在唯一性等问题·　证明该控制问题中的价值函数是这一方程的一个粘性解·     

由Levy过程驱动的倒向随机微分方程在随机单调条件下解的存在唯一性

本文讨论了如下的由Levy过程驱动的倒向随机微分方程适应解的存在唯一性■其中W_s是一Wiener过程,H_s为由Levy过程构成Teugels鞅.我们通过构造函数逼近序列的方法证明了,在漂移系数f关于Y满足随机单调,f关于Z和U满足随机Lipschitz条件下,方程存在唯一适应解.     

Maximum principle for anticipated recursive stochastic optimal control problem with delay and Lvy processes

In this paper,we study the stochastic maximum principle for optimal control problem of anticipated forward-backward system with delay and Lvy processes as the random disturbance. This control system can be described by the anticipated forward-backward stochastic differential equations with delay and L′evy processes(AFBSDEDLs),we first obtain the existence and uniqueness theorem of adapted solutions for AFBSDEDLs; combining the AFBSDEDLs' preliminary result with certain classical convex variational techniques,the corresponding maximum principle is proved.     

UNIQUENESS OF VISCOSITY SOLUTIONS OF STOCHASTIC HAMILTON-JACOBI EQUATIONS

This article is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi(HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the associated stochastic HJ equation.     

Optimal control of random evolutions

We consider a stochastic control problem for a random evolution. We study the Bellman equation of the problem and we prove the existence of an optimal stochastic control which is Markovian. This problem enables us to approximate the general problem of the optimal control of solutions of stochastic differential equations.     

An interval approach based on expectation optimization for fuzzy random bilevel linear programming problems

This paper considers a class of bilevel linear programming problems in which the coefficients of both objective functions are fuzzy random variables. The main idea of this paper is to introduce the Pareto optimal solution in a multi-objective bilevel programming problem as a solution for a fuzzy random bilevel programming problem. To this end, a stochastic interval bilevel linear programming problem is first introduced in terms of α-cuts of fuzzy random variables. On the basis of an order relation of interval numbers and the expectation optimization model, the stochastic interval bilevel linear programming problem can be transformed into a multi-objective bilevel programming problem which is solved by means of weighted linear combination technique. In order to compare different optimal solutions depending on different cuts, two criterions are given to provide the preferable optimal solutions for the upper and lower level decision makers respectively. Finally, a production planning problem is given to demonstrate the feasibility of the proposed approach.     

Continuous-Time Mean-Variance Portfolio Selection with Random Horizon

This paper examines the continuous-time mean-variance optimal portfolio selection problem with random market parameters and random time horizon. Treating this problem as a linearly constrained stochastic linear-quadratic optimal control problem, I explicitly derive the efficient portfolios and efficient frontier in closed forms based on the solutions of two backward stochastic differential equations. Some related issues such as a minimum variance portfolio and a mutual fund theorem are also addressed. All the results are markedly different from those in the problem with deterministic exit time. A key part of my analysis involves proving the global solvability of a stochastic Riccati equation, which is interesting in its own right.     

Optimal Control of Point Processes with Noisy Observations: The Maximum Principle

This paper studies the optimal control problem for point processes with Gaussian white-noised observations. A general maximum principle is proved for the partially observed optimal control of point processes, without using the associated filtering equation . Adjoint flows—the adjoint processes of the stochastic flows of the optimal system—are introduced, and their relations are established. Adjoint vector fields , which are observation-predictable, are introduced as the solutions of associated backward stochastic integral-partial differential equtions driven by the observation process. In a heuristic way, their relations are explained, and the adjoint processes are expressed in terms of the adjoint vector fields, their gradients and Hessians, along the optimal state process. In this way the adjoint processes are naturally connected to the adjoint equation of the associated filtering equation . This shows that the conditional expectation in the maximum condition is computable through filtering the optimal state, as usually expected. Some variants of the partially observed stochastic maximum principle are derived, and the corresponding maximum conditions are quite different from the counterpart for the diffusion case. Finally, as an example, a quadratic optimal control problem with a free Poisson process and a Gaussian white-noised observation is explicitly solved using the partially observed maximum principle. Accepted 8 August 2001. Online publication 17 December, 2001.     

Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management

We formulate and investigate a general stochastic control problem under a progressive enlargement of filtration. The global information is enlarged from a reference filtration and the knowledge of multiple random times together with associated marks when they occur. By working under a density hypothesis on the conditional joint distribution of the random times and marks, we prove a decomposition of the original stochastic control problem under the global filtration into classical stochastic control problems under the reference filtration, which is determined in a finite backward induction. Our method revisits and extends in particular stochastic control of diffusion processes with a finite number of jumps. This study is motivated by optimization problems arising in default risk management, and we provide applications of our decomposition result for the indifference pricing of defaultable claims, and the optimal investment under bilateral counterparty risk. The solutions are expressed in terms of BSDEs involving only Brownian filtration, and remarkably without jump terms coming from the default times and marks in the global filtration.     

Averaging evolutionary equations perturbed by random processes with jumps

Weak convergence of measures generated by solutions of an evolutionary equation dependent on a small parameter to the unique solution of the martingale problem corresponding to the stochastic evolutionary equation is proved. The coefficients of the initial equation depend on random Markov processes with jumps.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 197–207, February, 1992.     

Models and model value in stochastic programming

Finding optimal decisions often involves the consideration of certain random or unknown parameters. A standard approach is to replace the random parameters by the expectations and to solve a deterministic mathematical program. A second approach is to consider possible future scenarios and the decision that would be best under each of these scenarios. The question then becomes how to choose among these alternatives. Both approaches may produce solutions that are far from optimal in the stochastic programming model that explicitly includes the random parameters. In this paper, we illustrate this advantage of a stochastic program model through two examples that are representative of the range of problems considered in stochastic programming. The paper focuses on the relative value of the stochastic program solution over a deterministic problem solution.The author's work was supported in part by the National Science Foundation under Grant DDM-9215921.