由Levy过程驱动的随机线性二次最优控制问题及含Markov链的倒向随机微分方程
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摘要
在这篇论文中,我们以二次最优控制问题和倒向随机微分方程的理论为基础,研究由Lévy过程驱动的随机线性二次最优控制问题及含马尔科夫链的倒向随机微分方程。本文共由四章组成。
在第一章中,我们简单的回顾了本文所研究问题的历史文献,并列出这篇论文所得到的主要结果。
在第二章和第三章中,我们考虑由Lévy过程驱动的随机线性二次最优控制问题。众所周知,最优控制理论在实际生活中有着重要且广泛的应用。其中,线性二次最优控制问题是最优控制问题中应用最广泛的问题之一。在其形式首次被Kalman[22]提出之后,关于该问题的理论得到了很好的发展。用Brown运动刻画随机干扰的随机二次最优控制问题由Wonham([53,54])首次提出,而后得到了各个领域的研究专家深入广泛的研究,相关文献见[13,5,46,59,45]及其参考文献。另外,Lévy过程是一类有独立平稳增量及右连左极轨道的过程。因为该过程在金融及其它领域有许多的应用,所以受到了众多的关注和研究。在第二章中,我们考虑一类有限时间区间内用Lévy过程描述随机干扰的随机线性二次最优控制问题。我们证明了如果一类黎卡提方程有解,则该线性二次最优控制问题适定并且可以根据此黎卡提方程的解构造最优状态反馈控制。另外,我们还给出了这类黎卡提方程可解的一些充分条件。
第三章的内容是第二章内容的延续。我们知道,无穷时间区间内用Brown运动刻画随机干扰的线性二次最优控制已经得到广泛且深入的研究,相关文献见[59],[46],[47],[55]及其参考文献。在第三章中,我们考虑一类无穷时间区间内用Lévy过程描述随机干扰的随机线性二次最优控制问题。我们给出了这类二次最优控制问题的可达性和一类代数黎卡提方程的稳定解之间的联系。另外,我们还利用一类半定规划及其对偶问题来研究这类控制问题的最优控制。
在本文的最后一章中,我们研究了含马尔科夫链的倒向随机微分方程和一类偏微分方程的均匀化。对倒向随机微分方程的研究起源于对随机控制的研究([6]),Pardoux和Peng[37]首次给出一般形式的倒向随机微分方程的结果。此后,由于倒向随机微分方程与数学金融,随机控制和偏微分方程之间的联系,该类方程已经得到了广泛的研究,相关文献见[23,14,15]及其参考文献。值得注意的是倒向随机微分方程为我们研究偏微分方程的均匀化提供了一种概率工具。偏微分方程的均匀化是指这样一种过程:用新的系数代替变化较快的的系数,并且取代后得到的方程的解和原方程的解相近。在第四章中,受到一类含马尔科夫跳的随机线性二次最优控制问题的启发,我们考虑一类系数受马尔科夫链干扰的倒向随机微分方程,并且证明该方程存在唯一解。当该马尔科夫链的状态空间很大时,为了区分不同状态间快和慢的转移速率,我们知道可以引入一个小参数(ε＞0),从而认为该马尔科夫链为含双时间尺度的受奇异干扰的马尔科夫链。利用奇异扰动技巧和概率逼近的方法,可以得到当ε→0时,受奇异干扰的马尔科夫链的渐近概率分布。更多的细节可参照[61]。在第四章中,利用这类受奇异干扰的马尔科夫链的渐近概率分布,我们给出了系数含受奇异干扰的马尔科夫链的倒向随机微分方程弱收敛的结果。最后,基于倒向随机微分方程和偏微分方程的联系,我们给出一类偏微分方程的均匀化结果。
This dissertation is developed on the theory of linear quadratic(LQ in short) optimal control problem and backward stochastic differential equations(BSDEs in short).And we mainly focus on stochastic LQ control problem driven by Lévy processes and BSDEs with Markov chains.This thesis includes four chapters.
In Chapter 1,a brief review of the historical literature in these topics we concerned is given.And also,we present the main results obtained in this thesis.
In Chapter 2 and Chapter 3,we will consider stochastic LQ control problem driven by Lévy processes both in finite time horizon and in infinite time horizon.On one hand, as we all know,optimal control theory has great applications in practical problems. Especially,the LQ optimal control problem is one of the most popular problem in optimal control theory.After presented firstly by Kalman[22],the theory of such problem has been well established.Then,stochastic LQ problem with Brownian motion as the noise source in finite time horizon,which was firstly given by Wonham[53,54], has been studied extensively by many researchers,such as[13,5,46,59,45]and their references.On another hand,Lévy process,which has independent,stationary increments and càdlàg trajectories,has received much attention in recent years because of its many applications in finance and other fields.So we consider stochastic LQ control problem driven by Lévy processes in finite time horizon in Chapter 2.And we will show that,if one kind of Riccati equation admits a solution,this LQ control problem is well-posed and optimal feedback controls can be obtained via the solution. We also give some sufficient conditions for the solvability of this Riccati equation.
Chapter 3 is a continuation of Chapter 2,in which we will discuss stochastic LQ control problem driven by Lévy processes in infinite time horizon.As we know,for infinite time horizon,stochastic LQ control problem with the system using Brownian motion to describe the noise was widely studied,such as in[59],[46],[47],[55]and their references.In this chapter,we will give the relation between attainability of this LQ problem and the stabilizing solution of one kind of algebraic Riccati equation.And we also study the attainability of this LQ problem via a semi-definite programming (SDP in short) and its dual problem.
The last chapter is devoted to the study of BSDEs with Markov chains and homogenization of systems of partial differential equations(PDEs in short).The study of BSDEs stems from the research about stochastic control problem([6]).After the pioneering work of Pardoux and Peng[37]about BSDEs in a general case,BSDEs have been extensively studied recently because of its connections with mathematical finance, stochastic control and PDEs,see among[23,14,15].It is noted that BSDEs provide a probabilistic tool to study the homogenization of PDEs,which is the process of replacing rapidly varying coefficients by new ones thus the solutions are close.In Chapter 4, motivated by a stochastic LQ control problem with Markov jumps,we consider BSDEs with coefficients disturbed by Markov chains and give the uniqueness and existence results about the solution.When the Markov chain has a large state space,to distinguish the fast transitions from the slow transitions among different states,it is known that a small parameter(ε＞0) can be introduced which leads to a singularly perturbed Markov chain involving two time scales,namely,the actual time t and the stretched time t/εThen,using singular perturbation techniques and probabilistic approaches, the asymptotic probability distributions of singularly perturbed Markov chains can be derived asε→0.More details can be found in[61].In this chapter,based on the asymptotic probability distributions,we will consider the weak convergence of BSDEs with singularly perturbed Markov chains.We also give the homogenization result of one kind of PDE based on the connection between BSDEs and PDEs.
在第一章中,我们简单的回顾了本文所研究问题的历史文献,并列出这篇论文所得到的主要结果。
在第二章和第三章中,我们考虑由Lévy过程驱动的随机线性二次最优控制问题。众所周知,最优控制理论在实际生活中有着重要且广泛的应用。其中,线性二次最优控制问题是最优控制问题中应用最广泛的问题之一。在其形式首次被Kalman[22]提出之后,关于该问题的理论得到了很好的发展。用Brown运动刻画随机干扰的随机二次最优控制问题由Wonham([53,54])首次提出,而后得到了各个领域的研究专家深入广泛的研究,相关文献见[13,5,46,59,45]及其参考文献。另外,Lévy过程是一类有独立平稳增量及右连左极轨道的过程。因为该过程在金融及其它领域有许多的应用,所以受到了众多的关注和研究。在第二章中,我们考虑一类有限时间区间内用Lévy过程描述随机干扰的随机线性二次最优控制问题。我们证明了如果一类黎卡提方程有解,则该线性二次最优控制问题适定并且可以根据此黎卡提方程的解构造最优状态反馈控制。另外,我们还给出了这类黎卡提方程可解的一些充分条件。
第三章的内容是第二章内容的延续。我们知道,无穷时间区间内用Brown运动刻画随机干扰的线性二次最优控制已经得到广泛且深入的研究,相关文献见[59],[46],[47],[55]及其参考文献。在第三章中,我们考虑一类无穷时间区间内用Lévy过程描述随机干扰的随机线性二次最优控制问题。我们给出了这类二次最优控制问题的可达性和一类代数黎卡提方程的稳定解之间的联系。另外,我们还利用一类半定规划及其对偶问题来研究这类控制问题的最优控制。
在本文的最后一章中,我们研究了含马尔科夫链的倒向随机微分方程和一类偏微分方程的均匀化。对倒向随机微分方程的研究起源于对随机控制的研究([6]),Pardoux和Peng[37]首次给出一般形式的倒向随机微分方程的结果。此后,由于倒向随机微分方程与数学金融,随机控制和偏微分方程之间的联系,该类方程已经得到了广泛的研究,相关文献见[23,14,15]及其参考文献。值得注意的是倒向随机微分方程为我们研究偏微分方程的均匀化提供了一种概率工具。偏微分方程的均匀化是指这样一种过程:用新的系数代替变化较快的的系数,并且取代后得到的方程的解和原方程的解相近。在第四章中,受到一类含马尔科夫跳的随机线性二次最优控制问题的启发,我们考虑一类系数受马尔科夫链干扰的倒向随机微分方程,并且证明该方程存在唯一解。当该马尔科夫链的状态空间很大时,为了区分不同状态间快和慢的转移速率,我们知道可以引入一个小参数(ε＞0),从而认为该马尔科夫链为含双时间尺度的受奇异干扰的马尔科夫链。利用奇异扰动技巧和概率逼近的方法,可以得到当ε→0时,受奇异干扰的马尔科夫链的渐近概率分布。更多的细节可参照[61]。在第四章中,利用这类受奇异干扰的马尔科夫链的渐近概率分布,我们给出了系数含受奇异干扰的马尔科夫链的倒向随机微分方程弱收敛的结果。最后,基于倒向随机微分方程和偏微分方程的联系,我们给出一类偏微分方程的均匀化结果。
This dissertation is developed on the theory of linear quadratic(LQ in short) optimal control problem and backward stochastic differential equations(BSDEs in short).And we mainly focus on stochastic LQ control problem driven by Lévy processes and BSDEs with Markov chains.This thesis includes four chapters.
In Chapter 1,a brief review of the historical literature in these topics we concerned is given.And also,we present the main results obtained in this thesis.
In Chapter 2 and Chapter 3,we will consider stochastic LQ control problem driven by Lévy processes both in finite time horizon and in infinite time horizon.On one hand, as we all know,optimal control theory has great applications in practical problems. Especially,the LQ optimal control problem is one of the most popular problem in optimal control theory.After presented firstly by Kalman[22],the theory of such problem has been well established.Then,stochastic LQ problem with Brownian motion as the noise source in finite time horizon,which was firstly given by Wonham[53,54], has been studied extensively by many researchers,such as[13,5,46,59,45]and their references.On another hand,Lévy process,which has independent,stationary increments and càdlàg trajectories,has received much attention in recent years because of its many applications in finance and other fields.So we consider stochastic LQ control problem driven by Lévy processes in finite time horizon in Chapter 2.And we will show that,if one kind of Riccati equation admits a solution,this LQ control problem is well-posed and optimal feedback controls can be obtained via the solution. We also give some sufficient conditions for the solvability of this Riccati equation.
Chapter 3 is a continuation of Chapter 2,in which we will discuss stochastic LQ control problem driven by Lévy processes in infinite time horizon.As we know,for infinite time horizon,stochastic LQ control problem with the system using Brownian motion to describe the noise was widely studied,such as in[59],[46],[47],[55]and their references.In this chapter,we will give the relation between attainability of this LQ problem and the stabilizing solution of one kind of algebraic Riccati equation.And we also study the attainability of this LQ problem via a semi-definite programming (SDP in short) and its dual problem.
The last chapter is devoted to the study of BSDEs with Markov chains and homogenization of systems of partial differential equations(PDEs in short).The study of BSDEs stems from the research about stochastic control problem([6]).After the pioneering work of Pardoux and Peng[37]about BSDEs in a general case,BSDEs have been extensively studied recently because of its connections with mathematical finance, stochastic control and PDEs,see among[23,14,15].It is noted that BSDEs provide a probabilistic tool to study the homogenization of PDEs,which is the process of replacing rapidly varying coefficients by new ones thus the solutions are close.In Chapter 4, motivated by a stochastic LQ control problem with Markov jumps,we consider BSDEs with coefficients disturbed by Markov chains and give the uniqueness and existence results about the solution.When the Markov chain has a large state space,to distinguish the fast transitions from the slow transitions among different states,it is known that a small parameter(ε＞0) can be introduced which leads to a singularly perturbed Markov chain involving two time scales,namely,the actual time t and the stretched time t/εThen,using singular perturbation techniques and probabilistic approaches, the asymptotic probability distributions of singularly perturbed Markov chains can be derived asε→0.More details can be found in[61].In this chapter,based on the asymptotic probability distributions,we will consider the weak convergence of BSDEs with singularly perturbed Markov chains.We also give the homogenization result of one kind of PDE based on the connection between BSDEs and PDEs.
引文
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