带奇异摄动马氏链的倒向随机微分方程及其应用
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摘要
本篇论文主要讨论带奇异摄动马氏链的倒向随机微分方程(BSDEs)及相应偏微分方程(PDEs)的渐进性质和在随机控制及金融数学中的应用.论文包括以下三个部分：第一部分研究带奇异摄动马氏链的倒向随机微分方程在Meyer-Zheng拓扑下的弱收敛问题；第二部分研究随机体制转换系统的最优转换问题,通过带马氏链的斜反射倒向随机微分方程得到转换问题的最优解,并在奇异摄动马氏链情形下,得到相应变分不等式的渐进性质；第三部分给出带马氏链的倒向随机微分方程在随机控制中的一个应用,即正倒向体制转换系统的随机最大值原理.
本文中奇异摄动马氏链指转移速率为多时间尺度的马氏链.在许多物理模型中,系统中的组成元素以不同的速率发生变化,为了定量的研究这种变化,并且降低模型的复杂度,我们运用奇异摄动的方法,即假设马氏链的转移速率为两个时间尺度,进而把与快速变换相关的变量平均化,得到与平稳分布相关的极限问题.具体来说,我们假设马氏链at=αtε的生成矩阵为Qε(t)＝Q(t)/ε＋Q(t).这里,ε>0是一个时间参数,并且Q(t)和Q(t)都是一个马氏链的转移矩阵,其中Q(t)表示快速转换部分,Q(t)表示慢速部分.我们将研究在ε→0时,相应方程及优化控制问题的渐进性质.通过得到简化的极限问题,可以降低问题的计算复杂度.
下面将进一步介绍论文的内容及结构.
第一章,介绍本文中所研究问题的背景及基础知识.
第二章,我们研究了奇异摄动马氏链的倒向随机微分方程在Meyer-Zheng拓扑下的弱收敛问题.我们首先通过估计得到带参数ε的一族倒向随机微分方程的紧性,然后通过鞅问题刻画了极限过程.通过倒向随机微分方程给出的概率表示,我们得到相应偏微分方程在ε→0时的收敛性,并给出了数值计算例子.本章主要来自于：
R. Tao, Z. Wu and Q. Zhang, BSDEs with regime switching:Weak convergence and applications, Journal of Mathematical Analysis and Applications,407(1),97-111,2013.
第三章,我们研究了体制转换系统的最优转换问题,其中体制转换系统由带马氏链αt的随机微分方程描述,决策者可以在有限的控制集I={1,2，...,N}中转换状态以最大化效用函数.此问题存在两个“状态转换”,at的状态转换由市场确定,另一个由决策者根据市场体制及市场价格决定.我们通过带马氏链的斜反射倒向随机微分方程刻画了值函数,并构造出最优的转换策略.在马氏链的转移速率为两尺度的情形下,我们通过倒向随机微分方程方法证明了相应变分不等式的收敛性,从而证明了值函数的收敛性.我们也给出了一个数值计算的例子.本章主要来自于：
R. Tao, Z. Wu and Q. Zhang, Optimal switching under a regime switching model with two-time-scale Markov chains,已投稿.
第四章,我们研究了正倒向随机控制系统的最大值原理.假设系统由带马氏链的耦合的正倒向随机控制方程描述,控制域为凸集,通过凸变分方法,我们给出了最优控制存在的必要和充分条件,并给出一个在投资消费问题中的应用.本章主要来自于：
R. Tao and Z. Wu, Maximum principle for optimal control problems of forward-backward regime-switching system and applications, System and Control Letters,61(9),911-917,2012.
下面我们给出本论文的主要结论.1.带奇异摄动马氏链的倒向随机微分方程弱收敛问题
我们的目的是研究如下倒向随机微分方程的弱收敛：其中,马氏链αε(t)的生成矩阵为马氏链的状态空间为M=M1∪…∪Ml,其中Mk={Skl,…,skmk},这里K∈{1,...,L}, M=m1+…+ml.另外,Q(t)有对角线结构其中,(?)k∈{1,…l},Qk(t)是一个状态空间是Mk的马氏链的生成矩阵.
Xεt是带奇异摄动马氏链αεt的随机微分方程的解
马氏链αt或者它的生成矩阵Q(t)称为弱不可约,如果有唯一的非负解v(t)=(v1(t),...,vm0(t)).此解称为马氏链αt或者它的生成矩阵Q(t)的拟平稳分布.
这一部分的主要结果是如下定理：
定理0.1.Yεt是BSDEs(0.0.1)的解.在假设条件似2.3-A2.7)下,随机过程Yεt弱收敛于过程Yt,其中Yt是BSDE的解.这里,Bt是一个布朗运动,Vt(j)是马氏链αt的补偿鞅测度.αt的生成矩阵为Q(t)=diag(v1(t),...,v1(t))Q(t)diag(1m1,...,1ml).这里,1mk.=(1,...,1)'∈Rmk是一个mk维列向量,vk(t)=(v1k(t),…,vmkk(t))∈R1×mk是QK(t)的拟平稳分布..f的定义为F(t,i,x,y)=∑jmi=1v3i(t)f(t,sij,x,y).
为证明此定理,首先根据Meyer-Zheng判定准则得到Yεt的紧性.为了刻画极限过程,考虑如下算子其中这里Q(t)ij=(λij(t)),b,a的定义为通过算子g对应鞅问题解的唯一性,我们可以刻画Yεt的极限过程.
然后,我们分别在粘性解和经典解意义下,通过BSDE(0.0.1)给出对应耦合PDE方程组的概率表示,进而得到PDE方程组的收敛.这部分的主要结果是：
定理0.2.uε是如下反应扩散方程的唯一粘性解
则Vt∈[0,T]和x∈Rn,当ε→0,uε(t,x)收敛于u(t,x),其中u是下列方程唯一的粘性解
注0.1.uε(t,x)收敛于u(t,x)表示对于任给的(t,x)∈[0,T]×Rn和i∈Mk, uε(t,i,x)→u(t,k,x).
定理0.3.在假设条件(A2.3-A2.4)和(A2.6-A2.11)下,方程(0.0.3)有唯一的C1,2b解uε(>)t[0,T]和x∈Rn,当ε→0uε(t,x)收敛于u(t,x),这里u是极限方程(0.0.4)的唯一C1,2b解.
2.带奇异摄动马氏链的随机系统的最优转换问题及其渐进性质在这一部分,我们主要研究如下体制转换系统的最优转换问题,其中α(s)是连续时间有限状态时齐的马氏链.
决策者可以在有限的转换控制集N={1,...,N}中选择,即一个转换控制过程是一列(τn,ξn)n≥1,其中τn是一列停时,表示转换的时间,ξn是一列取值于N的随机变量,表示转换之后的状态.给定初始时间t和初始状态i,一个转换控制过程可以表示为这里1是一个示性函数.
我们的目标是选择一个转换控制过程Iti*来最大化效用函数
其中gij是从状态i变换到j的固定成本.Vi,p(t,x):=J(i,t,p,x,It,i,*)称为最优转换问题的值函数.
为解决此问题,我们运用BSDE方法.考虑如下带马氏链的斜反射倒向随机微分方程：
通过惩罚函数方法,BSDE (0.0.6)解的存在性结果由如下定理得到：
定理0.4.假设(A3.1-A3.3)成立,那么,BSDE (0.0.6)存在一个解(Yt,p,x,Zt,p,x,Wt,p,x,Kt,p,x)∈S2×M2×H2×N2.
下面,我们由验证定理来说明BSDE (0.0.6)解的唯一性.对任意转换控制过程我们定义如下增过程4I：引入下列带转换控制的BSDE:上述BSDE的唯一解记为(YiI,ZiI,WiI).
BSDE (0.0.6)解的唯一性及最优策略的选择可由下列验证定理得到：
定理0.5.假设(A3.1-A3.3)成立.设(Yt,p,xZt,p,x,Wt,p,x,Kt,p,x)是BSDE (0.0.6)在S2×M2×H2×N2中的一个解.那么,(1)任给I∈Ait,我们有Yit,p,x(s)≥YiI(s), s∈[t,T].
(2)令τ0*=t,ξ0=i.定义下列序列{τj*,ξj*}:其中ξ*j是一个随机变量使得那么,是最优转换问题的最优策略,并且
另外,我们有Yit,p,x(s)=YiI*(s),s∈[t,T],从而说明了BSDE(0.0.6)解的唯一性.
进一步,通过以下定理,我们给出最优转换问题的值函数对相应变分不等式的概率表示：
定理0.6.假设(A3.1-A3.3)成立.最优转换问题的值函数V(t，x)是如下变分不等式系统的唯一粘性解终端条件为Vi,p(T,x)=Φ(x),
下一部分,我们假设马氏链αε的结构为双时间尺度,即生成矩阵为αε的状态空间为M=M1∪…∪ML,其中Mk={sk1...,skmk},M=m1+…mL.生成矩阵有与第一部分相同的对角线结构.我们将研究当ε→0时最优转换问题对应变分不等式的收敛.
首先,我们给出关于增过程Kit,p,x的一个估计：
引理0.1.Kit,p,x对Lebesgue测度绝对连续,并且
考虑如下带参数ε的变分不等式系统：
下面,对于k=1,...,L,令Vi.skl(t,x)=Vi,k(t,x).定义一个平均系数的极限最优转换问题.vk=(v1k,...,vmkk)记为Qk的平稳分布.令记α为一个马氏链,生成矩阵为Q=diag(v1,...,vL)Qdiag(1m1,...,1mL),其中vk为Qk的平稳分布,1n=(1,...,1)'∈Rn.令Q=[λpq](p,q∈{1,...,L}).
考虑系数为b,σσ',f,马氏链为α的最优转换问题.对应变分不等式系统为终端条件为Vi,k(T,x)=Φ(x)其中
本部分的主要结果是：
定理0.7.对于k=1,...,L和l=1,...,mk,我们有这里,Vik(t,x)是极限变分不等式系统(0.0.9)的唯一粘性解.
3.正倒向体制转换系统的随机最大值原理
这一部分,我们主要研究如下正倒向随机控制系统的最优控制问题：
其中,马氏链的状态空间为M={1,...,k}. Wt=(Wt(1),...,Wt(k)), nt=(nt(l),..., nt(k)),其中,nt(j)=lαt-≠j}λ(αt-,j).
记U为取值与凸集U的可行控制集并且满足E∫0t|ut|2dt定义如下效用函数
这里,l,h,r是确定性函数.我们的目标是在U中寻找最优控制来最大化效用函数.首先考虑最优控制存在的必要条件.
设u(·)是最优控制问题的一个最优控制,对应的系统记为(X(·),Y(·),Z(·),W(·)).设υ(·)是另一个控制过程(不一定取值与U)并且满足u(·)+v(·)∈u.因为控制域U为凸,我们有(?)0≤p≤1, uρ(·):=u(·)+pv(·)∈u.引入如下变分方程：
首先,我们可以得到关于变分不等式的如下引理：
引理0.2.假设(A4.1-A4.3)成立,如下变分不等式成立
定义如下汉密尔顿函数H：[0,T]×M×R×R×Rl×dMρ×U×R×R1×d×R: H(t,i,x,y,z,w,u,p,k,q)=(p, b(t, i, x, u))+(k,σ(t, i, x, u))-(g, g(t, i, x, y, z, wn, u))+l(t, i, x, y, z, wn, u),(0.0.14)这里,wn=(w(l)n(1),…,w(k)n(k)), n(j)=1{i≠j}λij.
引入对偶方程：
运用Ito公式,可以得到本节的主要结果
定理0.8(最大值原理).设u(·)是一个最优控制,(X(·),Y(·),Z(·),W(·))是相应系统方程的解.(P(·),K(·),Q(·))是对偶方程的唯一解.那么,Vv∈U,我们有Hu·(v-vt)≤0,a.e.,a.s..
在一定的凸性条件下,我们还可以得到最优控制存在的充分条件：
定理0.9.设(A4.1-44.3)成立.此外,我们假设h,r,H对于变量(X,Y,Z,u,W)是凹的(Concave),并且YT=Φ(XT)是YT=K(αT)XT这种特殊形式,这里K(i)是一个确定性函数.(P,Q,K,M)是对偶方程控制取u(·)时的唯一解.那么,u(·)是最优控制如果满足(0.0.16).
在论文的最后部分,我们给出此最大值原理在一个投资消费问题中的应用.
In this paper, we will discuss the asymptotic property and applications of the back-ward stochastic differential equations (BSDEs) with singular perturbed Markov chain and the corresponding PDEs. This paper consists of three parts. In the first part, we studies the weak convergence of the BSDEs with singular perturbed Markov chain under Meyer-Zheng topology. In the second part, we considers the optimal switching problem for regime-switching system. We obtain the optimal switching strategy by virtue of the oblique reflected BSDEs with Markov chain. Moreover, when the Markov chain has a two-time-scale structure, we studies the asymptotic property of the corresponding vari-ation inequalities. In the third part, we give an application of the BSDEs with Markov chain in stochastic maximum principle for forward backward stochastic system.
In this paper, the singular perturbed Markov chain refers to the multi-time-scale Markov chain. In many physical models, different elements in a large system evolve at different rates. Some of them vary rapidly and others change slowly. Naturally, one wants to describe the largeness and smallness in a quantitative way. To reduce the complexity involved, we use a singular perturbation approach based on a two-time-scale model. The main idea is to formulate the problem using a Markov chain with two-time-scale structure. Then the variables associated with the fast scale are "averaged out" and replaced by the corresponding stationary distributions. In view of these, we can take αt=αtε governed by the generator Qε(t)=Q(t)/ε+Q(t). Here ε>0is a time scale parameter and both Q(t) and Q(t) are generators of a Markov chain. In this paper, Q(t) indicates the fast part and Q(t) the slow part. We will discuss the asymptotic property of the equations and the optimal switching or control problems. We can reduce the computational complexity by dealing with the limit problem.
Next, we introduce the content and structure of this paper.
In the first chapter, we give the research background and some preliminaries.
In the second chapter, we studies the weak convergence of the BSDEs with singular perturbed Markov chain under Meyer-Zheng topology. First, we obtain the tightness of the parameterized BSDEs with ε by some classical BSDEs estimates, then we characterize the limit process by martingale problem. By virtue of the probabilistic representation by the BSDEs, we obtain the convergence of the corresponding PDE system when ε→0. Moreover, we give a numerical example to demonstrate the convergence. The content of this chapter is included in the following paper:
R. Tao, Z. Wu and Q. Zhang, BSDEs with regime switching:Weak convergence and applications, Journal of Mathematical Analysis and Applications,407(1),97-111,2013.
In the third chapter, we consider the optimal switching problem for the regime-switching system, which is described by stochastic differential equations modulated by an Markov chain. We can choose an switching control process to maximize the payoff function. There are two kinds of "switchings" in this paper. The switching of the Markov chain is determined by the market, while the other one is switching control which is chosen by the controller. We obtain the value function and the optimal strategy by means of oblique reflected BSDEs with Markov chain. When the Markov chain has the two-time-scale structure, we obtain the convergence of the corresponding variational inequalities by virtue of the BSDE method, which implies the convergence of the value function. We also give a numerical example. The results of this chapter are from the following paper:
R. Tao, Z. Wu and Q. Zhang, Optimal switching under a regime switching model with two-time-scale Markov chains. submitted.
In the fourth chapter, we discuss the maximum principle for the forward backward stochastic system. Assume the system follows an coupled forward backward stochastic differential equation modulated by an Markov chain and the control domain is convex. By the convex variation method, we give both the necessary and sufficient condition for the optimal control. Moreover, we give an application in an consumption-investment problem. The results of this chapter are included in the following paper:
R. Tao and Z. Wu, Maximum principle for optimal control problems of forward-backward regime-switching system and applications, System and Control Letters,61(9),911-917,2012.
In the following, we show the main results of this paper.1. Weak convergence of the BSDEs with singular perturbed Markov chain. We aim to obtain the weak convergence of the following BSDE:
where the generator of the Markov chain αε(t) is given by
The state space of αε(t) is given by M=M1∪…∪Ml, where Mk={ski,skmk}, for k=1,...,l and M=m1+…+ml. Moreover, Q(t) has a block diagonal structure
where for k∈{1,..., l}, Qk(t) is generator for a Markov chain with state space Mk·`Xtε is the solution of the following stochastic differential equation with Markov chain αtε A Markov chain at or the generator Q(t) is called weakly irreducible if the system of equations has a unique nonnegative solution v(t)=(v1(t),...,vm0(t)), which is called the quasi-stationary distribution. The main result of this part is the following theorem.
Theorem0.1. Let Ytε be the solutions of BSDEs (0.0.17). Under the assumptions (A2.3-A2.7), the processes Yte converge weakly to a process Yt, which satisfies the following BSDE
where Bt is a Brownian motion and Vti(j) is the compensated martingale measure related to a Markov chain αt. The generator of αt is Q(t)=diag(v1(t),..,vl(t))Q(t)diag(1m1,...,1m), where1mk=(1,...,1)'∈Rmk is an mk-dimensional column vector and vk(t)=(v1k(t),...,vmkk(t))∈R1×mk is the quasi-stationary distribution of Qk(t). f is defined by f(t,i,x,y)=∑j=1mi vji(t)f(t,sij,x,y).
In order to prove this theorem, we first obtain the tightness of Ytε by Meyer-Zheng criterion. To characterize the limit process, consider the following operator where and Q(t)i3=(λij(t)), b, a are defined by By virtue of the unique solution of the martingale problem of g, we can characterize the limit process.
Thereafter, we give the probabilistic interpretation for the corresponding PDE sys-tem both in the sense of viscosity solution and classical solution. Then. we can get the convergence of the PDE system.
The results of this part are given by the following theorems.
Theorem0.2. uε is the viscosity solution of the following reaction-diffusion equation
then fort∈[0, T] and x∈Rn, uε(t,x) converges to u(t,x) ε→0, where u is the unique viscosity solution of the following equation
Remark0.1. Note that the convergence of uε(t, x) to u(t, x) implies that for any (t, x)∈[0,T] x Rn and i∈Mk, uε(t,i,x)→u(t,k,x).
Theorem0.3. Under assumptions (A2.3-A2.4) and (A2.6-A2.11), the PDE system (0.0.19) has a unique Cb1,2solution uε. For all t∈[0, T] and x∈Rn, uε(t, x) converges to u(t, x) as ε→0, where u is the unique Cb1,2solution of the limit PDE (0.0.20).
2. Optimal switching problem for regime switching system with singular per-turbed Markov chain.
In this part, we studies the optimal switching problem for the following regime switching system where a(s) is a continuous-time finite-state Markov chain.
The controller can choose a switching control in the switching set N={1,..., N}. A switching control is a double sequence (τn,ξn)n≥1, where τn is an increasing sequence of stopping times taking values in [t,T], representing the decision on "when to switch" and ξn are random variable valued in N, representing the new value of the regime after time τn. Given an initial regime i at time t, we define switching control process
where1is the indicator function.
Our objective is to find an admissible switching control process Ii,*such that
where the payoff function J(i, t,p, x, Ii) is defined as Here, gij is the transition cost from i to j.Vi,p(t, x):=J(i,t,p,x,Ii,*) is called the value function of the optimal switching problem.
To solve this problem, we use the BSDE method. Consider the following oblique reflected BSDE with Markov chain
By virtue of penalization method, we obtain the existence of the solution.
Theorem0.4. Assume (A3.1-A3.3). Then, BSDE (0.0.21) has a unique solution (Yt,p,x,Zt,p,x,Wt,p,x,Kt,p,x)∈S2×M2×H2×N2.
Next, we use verification theorem to show the uniqueness of the solution. For any switching control process we define the following increasing process AI: Consider the BSDE with switching control, whose unique solution is denoted as (YiI,ZiI,WiI
The uniqueness of the solution of BSDE (0.0.21) and the optimal strategy can be obtained by the following theorem.
Theorem0.5. Assume (A3.1-A3.3). Let (Yt,p,xZt,p,xWt,p,x,Kt,p,x) be a solution of BSDE (0.0.21) in S2×M2×H2×N2. Then,(1) For any I∈Ati, we have (2)Set T*0=t and ξ*0=i.Define a sequence {Tj*,ξj*}as follows: and ξj*is the random uariable such that Then. is an optimal strategy for the optimal switching problem and Moreover,we have which implies the uniqueness of the solution for BSDE(0.0.21).
Next,we give a probabilistic representation for the corresponding variational in-equalities.
Theorem0.6.Assume(A3.1-A3.3).The value functio V(t,x)is the unique viscosity solution of the following variational inequalities with Vi,p(T,x)=Φ(x),
In the next part,we assume that the Markov chain αε has a two-time-scale structure, which is generated by Qε=[λpqε]such that
where Q=[λpq]and Q=[λpq].Assume further that the state space of αε is given by M=M1∪…∪ML,where Mk={sk1,…,skmk}for k=1,…L and M=m1+…+mL.
First,we give a estimate of the increasing process Kit,p,x in the BSDE(0.0.21).
Lemma0.1. The process Kit,p,x is absolutely continuous with respect to the Lebesgue measure. Moreover, we have
Consider the following variational inequalities with parameter s:
We next define a limit optimal switching problem with averaged coefficients. Let vk=(v1k,...,vmkk) be the stationary distribution of Qk. We define Let α denote a new Markov chain generated by where vk is the stationary distribution of Qk and1n=(1,...,1)'∈Rn,Let Q=[λpq](p,q∈{1,...,L}.
Consider the limit optimal switching problem with coefficients b, σσ, f and the Markov chain α. The corresponding HJB equation is
with terminal condition Vi,k(T,x)=Φ(x), where
The main result of this part is the following theorem.
Theorem0.7. For k=1,...,L and l=l,...,mk, we have Vi,sklε(t,x)→Vik(t,x). Moreover, Vik(t,x) is the unique viscosity solution of the HJB equation (0.0.24) for the limit optimal switching problem.
3. Stochastic maximum principle for forward backward regime switching sys-tem.
In this part, we discuss the optimal control problem for the following regime switch-ing system where the state space of the Markov chain at is given by M={1,...,k}. Wt=(Wt(1),..., Wt(k)), nt=(nt(1),..., nt(k)),where nt(j)=1{αt-≠j}λ(αt-,j).
Denote μ the class of admissible control taking values in convex domain U and satisfying Define the cost functional as follows,
where l, h, r are deterministic measurable functions. The objective of our optimal control problem is to maximize the cost functional over the admissible control set U.
First, we consider the necessary condition of the existence of the optimal control.
Let u(-) be an optimal control for the control problem (0.0.25), the corresponding trajectory is denoted by (x(·), y(·), z(·), W(·)). Let v(·) be another adapted control process (need not take values in U) satisfying u(·)+v(·)∈U. For the reason that the control domain U is convex, we have for any0≤ρ<1, uρ(·):=u(·)+ρv(·)∈U.
We introduce the following variational equation, which is a linear FBSDE with Markov chain:
Then we can prove the following variational inequality:
Lemma0.2. Under the assumptions (A4-1-A4-3), the following variational inequality holds:
Define the Hamiltonian H:[0,T]×M×R×R×R1×d×Mρ×U×R×R1×d×R as follows: H(t,i,x,y,z,w,u,p,k,q)=(p, b(t, i, x,u))+(k,σ(t,i,x,u))-(q, g(t, i, x, y, z, wn, u))+l(t, i, x, y, z, wnt u),(0.0.29)
where Wn=(W(1)n(1),…, W(k)n(k)) and n(j)=1{i≠j}λij Next, we introduce the following adjoint equation: Using Ito's formula, we can obtain the main result of this part.
Theorem0.8(Maximum Principle). Letu(·) be an optimal control and (X(·), Y(·), Z(·), W(·) be the corresponding trajectory.(P(·),K(·),Q(·)) is the unique solution of adjoint equa-tion. Then, for any v∈U, we have Hu·(v-ut)≤0,a.e.,a.s..(0.0.31)
With additional concave conditions, we can obtain the sufficient condition for the existence of the optimal control.
Theorem0.9. Suppose (A4.1-A4.3) hold. In addition, we assume h, r, H are all concave with respect to (X, Y, Z, u, W) and YT—Φ(XT) is of the special form YT—K(αT)Xt, where K is a deterministic measurable function. Let (P, Q, K, M) be the solution of the adjoint equation with respect to the control u(·). Then u(·) is an optimal control if it satisfies (0.0.31).
In the last part, we apply the maximum principle to a consumption-investment problem.
本文中奇异摄动马氏链指转移速率为多时间尺度的马氏链.在许多物理模型中,系统中的组成元素以不同的速率发生变化,为了定量的研究这种变化,并且降低模型的复杂度,我们运用奇异摄动的方法,即假设马氏链的转移速率为两个时间尺度,进而把与快速变换相关的变量平均化,得到与平稳分布相关的极限问题.具体来说,我们假设马氏链at=αtε的生成矩阵为Qε(t)＝Q(t)/ε＋Q(t).这里,ε>0是一个时间参数,并且Q(t)和Q(t)都是一个马氏链的转移矩阵,其中Q(t)表示快速转换部分,Q(t)表示慢速部分.我们将研究在ε→0时,相应方程及优化控制问题的渐进性质.通过得到简化的极限问题,可以降低问题的计算复杂度.
下面将进一步介绍论文的内容及结构.
第一章,介绍本文中所研究问题的背景及基础知识.
第二章,我们研究了奇异摄动马氏链的倒向随机微分方程在Meyer-Zheng拓扑下的弱收敛问题.我们首先通过估计得到带参数ε的一族倒向随机微分方程的紧性,然后通过鞅问题刻画了极限过程.通过倒向随机微分方程给出的概率表示,我们得到相应偏微分方程在ε→0时的收敛性,并给出了数值计算例子.本章主要来自于：
R. Tao, Z. Wu and Q. Zhang, BSDEs with regime switching:Weak convergence and applications, Journal of Mathematical Analysis and Applications,407(1),97-111,2013.
第三章,我们研究了体制转换系统的最优转换问题,其中体制转换系统由带马氏链αt的随机微分方程描述,决策者可以在有限的控制集I={1,2，...,N}中转换状态以最大化效用函数.此问题存在两个“状态转换”,at的状态转换由市场确定,另一个由决策者根据市场体制及市场价格决定.我们通过带马氏链的斜反射倒向随机微分方程刻画了值函数,并构造出最优的转换策略.在马氏链的转移速率为两尺度的情形下,我们通过倒向随机微分方程方法证明了相应变分不等式的收敛性,从而证明了值函数的收敛性.我们也给出了一个数值计算的例子.本章主要来自于：
R. Tao, Z. Wu and Q. Zhang, Optimal switching under a regime switching model with two-time-scale Markov chains,已投稿.
第四章,我们研究了正倒向随机控制系统的最大值原理.假设系统由带马氏链的耦合的正倒向随机控制方程描述,控制域为凸集,通过凸变分方法,我们给出了最优控制存在的必要和充分条件,并给出一个在投资消费问题中的应用.本章主要来自于：
R. Tao and Z. Wu, Maximum principle for optimal control problems of forward-backward regime-switching system and applications, System and Control Letters,61(9),911-917,2012.
下面我们给出本论文的主要结论.1.带奇异摄动马氏链的倒向随机微分方程弱收敛问题
我们的目的是研究如下倒向随机微分方程的弱收敛：其中,马氏链αε(t)的生成矩阵为马氏链的状态空间为M=M1∪…∪Ml,其中Mk={Skl,…,skmk},这里K∈{1,...,L}, M=m1+…+ml.另外,Q(t)有对角线结构其中,(?)k∈{1,…l},Qk(t)是一个状态空间是Mk的马氏链的生成矩阵.
Xεt是带奇异摄动马氏链αεt的随机微分方程的解
马氏链αt或者它的生成矩阵Q(t)称为弱不可约,如果有唯一的非负解v(t)=(v1(t),...,vm0(t)).此解称为马氏链αt或者它的生成矩阵Q(t)的拟平稳分布.
这一部分的主要结果是如下定理：
定理0.1.Yεt是BSDEs(0.0.1)的解.在假设条件似2.3-A2.7)下,随机过程Yεt弱收敛于过程Yt,其中Yt是BSDE的解.这里,Bt是一个布朗运动,Vt(j)是马氏链αt的补偿鞅测度.αt的生成矩阵为Q(t)=diag(v1(t),...,v1(t))Q(t)diag(1m1,...,1ml).这里,1mk.=(1,...,1)'∈Rmk是一个mk维列向量,vk(t)=(v1k(t),…,vmkk(t))∈R1×mk是QK(t)的拟平稳分布..f的定义为F(t,i,x,y)=∑jmi=1v3i(t)f(t,sij,x,y).
为证明此定理,首先根据Meyer-Zheng判定准则得到Yεt的紧性.为了刻画极限过程,考虑如下算子其中这里Q(t)ij=(λij(t)),b,a的定义为通过算子g对应鞅问题解的唯一性,我们可以刻画Yεt的极限过程.
然后,我们分别在粘性解和经典解意义下,通过BSDE(0.0.1)给出对应耦合PDE方程组的概率表示,进而得到PDE方程组的收敛.这部分的主要结果是：
定理0.2.uε是如下反应扩散方程的唯一粘性解
则Vt∈[0,T]和x∈Rn,当ε→0,uε(t,x)收敛于u(t,x),其中u是下列方程唯一的粘性解
注0.1.uε(t,x)收敛于u(t,x)表示对于任给的(t,x)∈[0,T]×Rn和i∈Mk, uε(t,i,x)→u(t,k,x).
定理0.3.在假设条件(A2.3-A2.4)和(A2.6-A2.11)下,方程(0.0.3)有唯一的C1,2b解uε(>)t[0,T]和x∈Rn,当ε→0uε(t,x)收敛于u(t,x),这里u是极限方程(0.0.4)的唯一C1,2b解.
2.带奇异摄动马氏链的随机系统的最优转换问题及其渐进性质在这一部分,我们主要研究如下体制转换系统的最优转换问题,其中α(s)是连续时间有限状态时齐的马氏链.
决策者可以在有限的转换控制集N={1,...,N}中选择,即一个转换控制过程是一列(τn,ξn)n≥1,其中τn是一列停时,表示转换的时间,ξn是一列取值于N的随机变量,表示转换之后的状态.给定初始时间t和初始状态i,一个转换控制过程可以表示为这里1是一个示性函数.
我们的目标是选择一个转换控制过程Iti*来最大化效用函数
其中gij是从状态i变换到j的固定成本.Vi,p(t,x):=J(i,t,p,x,It,i,*)称为最优转换问题的值函数.
为解决此问题,我们运用BSDE方法.考虑如下带马氏链的斜反射倒向随机微分方程：
通过惩罚函数方法,BSDE (0.0.6)解的存在性结果由如下定理得到：
定理0.4.假设(A3.1-A3.3)成立,那么,BSDE (0.0.6)存在一个解(Yt,p,x,Zt,p,x,Wt,p,x,Kt,p,x)∈S2×M2×H2×N2.
下面,我们由验证定理来说明BSDE (0.0.6)解的唯一性.对任意转换控制过程我们定义如下增过程4I：引入下列带转换控制的BSDE:上述BSDE的唯一解记为(YiI,ZiI,WiI).
BSDE (0.0.6)解的唯一性及最优策略的选择可由下列验证定理得到：
定理0.5.假设(A3.1-A3.3)成立.设(Yt,p,xZt,p,x,Wt,p,x,Kt,p,x)是BSDE (0.0.6)在S2×M2×H2×N2中的一个解.那么,(1)任给I∈Ait,我们有Yit,p,x(s)≥YiI(s), s∈[t,T].
(2)令τ0*=t,ξ0=i.定义下列序列{τj*,ξj*}:其中ξ*j是一个随机变量使得那么,是最优转换问题的最优策略,并且
另外,我们有Yit,p,x(s)=YiI*(s),s∈[t,T],从而说明了BSDE(0.0.6)解的唯一性.
进一步,通过以下定理,我们给出最优转换问题的值函数对相应变分不等式的概率表示：
定理0.6.假设(A3.1-A3.3)成立.最优转换问题的值函数V(t，x)是如下变分不等式系统的唯一粘性解终端条件为Vi,p(T,x)=Φ(x),
下一部分,我们假设马氏链αε的结构为双时间尺度,即生成矩阵为αε的状态空间为M=M1∪…∪ML,其中Mk={sk1...,skmk},M=m1+…mL.生成矩阵有与第一部分相同的对角线结构.我们将研究当ε→0时最优转换问题对应变分不等式的收敛.
首先,我们给出关于增过程Kit,p,x的一个估计：
引理0.1.Kit,p,x对Lebesgue测度绝对连续,并且
考虑如下带参数ε的变分不等式系统：
下面,对于k=1,...,L,令Vi.skl(t,x)=Vi,k(t,x).定义一个平均系数的极限最优转换问题.vk=(v1k,...,vmkk)记为Qk的平稳分布.令记α为一个马氏链,生成矩阵为Q=diag(v1,...,vL)Qdiag(1m1,...,1mL),其中vk为Qk的平稳分布,1n=(1,...,1)'∈Rn.令Q=[λpq](p,q∈{1,...,L}).
考虑系数为b,σσ',f,马氏链为α的最优转换问题.对应变分不等式系统为终端条件为Vi,k(T,x)=Φ(x)其中
本部分的主要结果是：
定理0.7.对于k=1,...,L和l=1,...,mk,我们有这里,Vik(t,x)是极限变分不等式系统(0.0.9)的唯一粘性解.
3.正倒向体制转换系统的随机最大值原理
这一部分,我们主要研究如下正倒向随机控制系统的最优控制问题：
其中,马氏链的状态空间为M={1,...,k}. Wt=(Wt(1),...,Wt(k)), nt=(nt(l),..., nt(k)),其中,nt(j)=lαt-≠j}λ(αt-,j).
记U为取值与凸集U的可行控制集并且满足E∫0t|ut|2dt定义如下效用函数
这里,l,h,r是确定性函数.我们的目标是在U中寻找最优控制来最大化效用函数.首先考虑最优控制存在的必要条件.
设u(·)是最优控制问题的一个最优控制,对应的系统记为(X(·),Y(·),Z(·),W(·)).设υ(·)是另一个控制过程(不一定取值与U)并且满足u(·)+v(·)∈u.因为控制域U为凸,我们有(?)0≤p≤1, uρ(·):=u(·)+pv(·)∈u.引入如下变分方程：
首先,我们可以得到关于变分不等式的如下引理：
引理0.2.假设(A4.1-A4.3)成立,如下变分不等式成立
定义如下汉密尔顿函数H：[0,T]×M×R×R×Rl×dMρ×U×R×R1×d×R: H(t,i,x,y,z,w,u,p,k,q)=(p, b(t, i, x, u))+(k,σ(t, i, x, u))-(g, g(t, i, x, y, z, wn, u))+l(t, i, x, y, z, wn, u),(0.0.14)这里,wn=(w(l)n(1),…,w(k)n(k)), n(j)=1{i≠j}λij.
引入对偶方程：
运用Ito公式,可以得到本节的主要结果
定理0.8(最大值原理).设u(·)是一个最优控制,(X(·),Y(·),Z(·),W(·))是相应系统方程的解.(P(·),K(·),Q(·))是对偶方程的唯一解.那么,Vv∈U,我们有Hu·(v-vt)≤0,a.e.,a.s..
在一定的凸性条件下,我们还可以得到最优控制存在的充分条件：
定理0.9.设(A4.1-44.3)成立.此外,我们假设h,r,H对于变量(X,Y,Z,u,W)是凹的(Concave),并且YT=Φ(XT)是YT=K(αT)XT这种特殊形式,这里K(i)是一个确定性函数.(P,Q,K,M)是对偶方程控制取u(·)时的唯一解.那么,u(·)是最优控制如果满足(0.0.16).
在论文的最后部分,我们给出此最大值原理在一个投资消费问题中的应用.
In this paper, we will discuss the asymptotic property and applications of the back-ward stochastic differential equations (BSDEs) with singular perturbed Markov chain and the corresponding PDEs. This paper consists of three parts. In the first part, we studies the weak convergence of the BSDEs with singular perturbed Markov chain under Meyer-Zheng topology. In the second part, we considers the optimal switching problem for regime-switching system. We obtain the optimal switching strategy by virtue of the oblique reflected BSDEs with Markov chain. Moreover, when the Markov chain has a two-time-scale structure, we studies the asymptotic property of the corresponding vari-ation inequalities. In the third part, we give an application of the BSDEs with Markov chain in stochastic maximum principle for forward backward stochastic system.
In this paper, the singular perturbed Markov chain refers to the multi-time-scale Markov chain. In many physical models, different elements in a large system evolve at different rates. Some of them vary rapidly and others change slowly. Naturally, one wants to describe the largeness and smallness in a quantitative way. To reduce the complexity involved, we use a singular perturbation approach based on a two-time-scale model. The main idea is to formulate the problem using a Markov chain with two-time-scale structure. Then the variables associated with the fast scale are "averaged out" and replaced by the corresponding stationary distributions. In view of these, we can take αt=αtε governed by the generator Qε(t)=Q(t)/ε+Q(t). Here ε>0is a time scale parameter and both Q(t) and Q(t) are generators of a Markov chain. In this paper, Q(t) indicates the fast part and Q(t) the slow part. We will discuss the asymptotic property of the equations and the optimal switching or control problems. We can reduce the computational complexity by dealing with the limit problem.
Next, we introduce the content and structure of this paper.
In the first chapter, we give the research background and some preliminaries.
In the second chapter, we studies the weak convergence of the BSDEs with singular perturbed Markov chain under Meyer-Zheng topology. First, we obtain the tightness of the parameterized BSDEs with ε by some classical BSDEs estimates, then we characterize the limit process by martingale problem. By virtue of the probabilistic representation by the BSDEs, we obtain the convergence of the corresponding PDE system when ε→0. Moreover, we give a numerical example to demonstrate the convergence. The content of this chapter is included in the following paper:
R. Tao, Z. Wu and Q. Zhang, BSDEs with regime switching:Weak convergence and applications, Journal of Mathematical Analysis and Applications,407(1),97-111,2013.
In the third chapter, we consider the optimal switching problem for the regime-switching system, which is described by stochastic differential equations modulated by an Markov chain. We can choose an switching control process to maximize the payoff function. There are two kinds of "switchings" in this paper. The switching of the Markov chain is determined by the market, while the other one is switching control which is chosen by the controller. We obtain the value function and the optimal strategy by means of oblique reflected BSDEs with Markov chain. When the Markov chain has the two-time-scale structure, we obtain the convergence of the corresponding variational inequalities by virtue of the BSDE method, which implies the convergence of the value function. We also give a numerical example. The results of this chapter are from the following paper:
R. Tao, Z. Wu and Q. Zhang, Optimal switching under a regime switching model with two-time-scale Markov chains. submitted.
In the fourth chapter, we discuss the maximum principle for the forward backward stochastic system. Assume the system follows an coupled forward backward stochastic differential equation modulated by an Markov chain and the control domain is convex. By the convex variation method, we give both the necessary and sufficient condition for the optimal control. Moreover, we give an application in an consumption-investment problem. The results of this chapter are included in the following paper:
R. Tao and Z. Wu, Maximum principle for optimal control problems of forward-backward regime-switching system and applications, System and Control Letters,61(9),911-917,2012.
In the following, we show the main results of this paper.1. Weak convergence of the BSDEs with singular perturbed Markov chain. We aim to obtain the weak convergence of the following BSDE:
where the generator of the Markov chain αε(t) is given by
The state space of αε(t) is given by M=M1∪…∪Ml, where Mk={ski,skmk}, for k=1,...,l and M=m1+…+ml. Moreover, Q(t) has a block diagonal structure
where for k∈{1,..., l}, Qk(t) is generator for a Markov chain with state space Mk·`Xtε is the solution of the following stochastic differential equation with Markov chain αtε A Markov chain at or the generator Q(t) is called weakly irreducible if the system of equations has a unique nonnegative solution v(t)=(v1(t),...,vm0(t)), which is called the quasi-stationary distribution. The main result of this part is the following theorem.
Theorem0.1. Let Ytε be the solutions of BSDEs (0.0.17). Under the assumptions (A2.3-A2.7), the processes Yte converge weakly to a process Yt, which satisfies the following BSDE
where Bt is a Brownian motion and Vti(j) is the compensated martingale measure related to a Markov chain αt. The generator of αt is Q(t)=diag(v1(t),..,vl(t))Q(t)diag(1m1,...,1m), where1mk=(1,...,1)'∈Rmk is an mk-dimensional column vector and vk(t)=(v1k(t),...,vmkk(t))∈R1×mk is the quasi-stationary distribution of Qk(t). f is defined by f(t,i,x,y)=∑j=1mi vji(t)f(t,sij,x,y).
In order to prove this theorem, we first obtain the tightness of Ytε by Meyer-Zheng criterion. To characterize the limit process, consider the following operator where and Q(t)i3=(λij(t)), b, a are defined by By virtue of the unique solution of the martingale problem of g, we can characterize the limit process.
Thereafter, we give the probabilistic interpretation for the corresponding PDE sys-tem both in the sense of viscosity solution and classical solution. Then. we can get the convergence of the PDE system.
The results of this part are given by the following theorems.
Theorem0.2. uε is the viscosity solution of the following reaction-diffusion equation
then fort∈[0, T] and x∈Rn, uε(t,x) converges to u(t,x) ε→0, where u is the unique viscosity solution of the following equation
Remark0.1. Note that the convergence of uε(t, x) to u(t, x) implies that for any (t, x)∈[0,T] x Rn and i∈Mk, uε(t,i,x)→u(t,k,x).
Theorem0.3. Under assumptions (A2.3-A2.4) and (A2.6-A2.11), the PDE system (0.0.19) has a unique Cb1,2solution uε. For all t∈[0, T] and x∈Rn, uε(t, x) converges to u(t, x) as ε→0, where u is the unique Cb1,2solution of the limit PDE (0.0.20).
2. Optimal switching problem for regime switching system with singular per-turbed Markov chain.
In this part, we studies the optimal switching problem for the following regime switching system where a(s) is a continuous-time finite-state Markov chain.
The controller can choose a switching control in the switching set N={1,..., N}. A switching control is a double sequence (τn,ξn)n≥1, where τn is an increasing sequence of stopping times taking values in [t,T], representing the decision on "when to switch" and ξn are random variable valued in N, representing the new value of the regime after time τn. Given an initial regime i at time t, we define switching control process
where1is the indicator function.
Our objective is to find an admissible switching control process Ii,*such that
where the payoff function J(i, t,p, x, Ii) is defined as Here, gij is the transition cost from i to j.Vi,p(t, x):=J(i,t,p,x,Ii,*) is called the value function of the optimal switching problem.
To solve this problem, we use the BSDE method. Consider the following oblique reflected BSDE with Markov chain
By virtue of penalization method, we obtain the existence of the solution.
Theorem0.4. Assume (A3.1-A3.3). Then, BSDE (0.0.21) has a unique solution (Yt,p,x,Zt,p,x,Wt,p,x,Kt,p,x)∈S2×M2×H2×N2.
Next, we use verification theorem to show the uniqueness of the solution. For any switching control process we define the following increasing process AI: Consider the BSDE with switching control, whose unique solution is denoted as (YiI,ZiI,WiI
The uniqueness of the solution of BSDE (0.0.21) and the optimal strategy can be obtained by the following theorem.
Theorem0.5. Assume (A3.1-A3.3). Let (Yt,p,xZt,p,xWt,p,x,Kt,p,x) be a solution of BSDE (0.0.21) in S2×M2×H2×N2. Then,(1) For any I∈Ati, we have (2)Set T*0=t and ξ*0=i.Define a sequence {Tj*,ξj*}as follows: and ξj*is the random uariable such that Then. is an optimal strategy for the optimal switching problem and Moreover,we have which implies the uniqueness of the solution for BSDE(0.0.21).
Next,we give a probabilistic representation for the corresponding variational in-equalities.
Theorem0.6.Assume(A3.1-A3.3).The value functio V(t,x)is the unique viscosity solution of the following variational inequalities with Vi,p(T,x)=Φ(x),
In the next part,we assume that the Markov chain αε has a two-time-scale structure, which is generated by Qε=[λpqε]such that
where Q=[λpq]and Q=[λpq].Assume further that the state space of αε is given by M=M1∪…∪ML,where Mk={sk1,…,skmk}for k=1,…L and M=m1+…+mL.
First,we give a estimate of the increasing process Kit,p,x in the BSDE(0.0.21).
Lemma0.1. The process Kit,p,x is absolutely continuous with respect to the Lebesgue measure. Moreover, we have
Consider the following variational inequalities with parameter s:
We next define a limit optimal switching problem with averaged coefficients. Let vk=(v1k,...,vmkk) be the stationary distribution of Qk. We define Let α denote a new Markov chain generated by where vk is the stationary distribution of Qk and1n=(1,...,1)'∈Rn,Let Q=[λpq](p,q∈{1,...,L}.
Consider the limit optimal switching problem with coefficients b, σσ, f and the Markov chain α. The corresponding HJB equation is
with terminal condition Vi,k(T,x)=Φ(x), where
The main result of this part is the following theorem.
Theorem0.7. For k=1,...,L and l=l,...,mk, we have Vi,sklε(t,x)→Vik(t,x). Moreover, Vik(t,x) is the unique viscosity solution of the HJB equation (0.0.24) for the limit optimal switching problem.
3. Stochastic maximum principle for forward backward regime switching sys-tem.
In this part, we discuss the optimal control problem for the following regime switch-ing system where the state space of the Markov chain at is given by M={1,...,k}. Wt=(Wt(1),..., Wt(k)), nt=(nt(1),..., nt(k)),where nt(j)=1{αt-≠j}λ(αt-,j).
Denote μ the class of admissible control taking values in convex domain U and satisfying Define the cost functional as follows,
where l, h, r are deterministic measurable functions. The objective of our optimal control problem is to maximize the cost functional over the admissible control set U.
First, we consider the necessary condition of the existence of the optimal control.
Let u(-) be an optimal control for the control problem (0.0.25), the corresponding trajectory is denoted by (x(·), y(·), z(·), W(·)). Let v(·) be another adapted control process (need not take values in U) satisfying u(·)+v(·)∈U. For the reason that the control domain U is convex, we have for any0≤ρ<1, uρ(·):=u(·)+ρv(·)∈U.
We introduce the following variational equation, which is a linear FBSDE with Markov chain:
Then we can prove the following variational inequality:
Lemma0.2. Under the assumptions (A4-1-A4-3), the following variational inequality holds:
Define the Hamiltonian H:[0,T]×M×R×R×R1×d×Mρ×U×R×R1×d×R as follows: H(t,i,x,y,z,w,u,p,k,q)=(p, b(t, i, x,u))+(k,σ(t,i,x,u))-(q, g(t, i, x, y, z, wn, u))+l(t, i, x, y, z, wnt u),(0.0.29)
where Wn=(W(1)n(1),…, W(k)n(k)) and n(j)=1{i≠j}λij Next, we introduce the following adjoint equation: Using Ito's formula, we can obtain the main result of this part.
Theorem0.8(Maximum Principle). Letu(·) be an optimal control and (X(·), Y(·), Z(·), W(·) be the corresponding trajectory.(P(·),K(·),Q(·)) is the unique solution of adjoint equa-tion. Then, for any v∈U, we have Hu·(v-ut)≤0,a.e.,a.s..(0.0.31)
With additional concave conditions, we can obtain the sufficient condition for the existence of the optimal control.
Theorem0.9. Suppose (A4.1-A4.3) hold. In addition, we assume h, r, H are all concave with respect to (X, Y, Z, u, W) and YT—Φ(XT) is of the special form YT—K(αT)Xt, where K is a deterministic measurable function. Let (P, Q, K, M) be the solution of the adjoint equation with respect to the control u(·). Then u(·) is an optimal control if it satisfies (0.0.31).
In the last part, we apply the maximum principle to a consumption-investment problem.
引文
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[42]S. Hamadene and M. Morlais, Viscosity Solutions of Systems of PDEs with Inter-Connected Obstacles and Switching Problem, Appl. Math. Optim.67 (2013) 163-196.
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[51]H. Kushner, F. Schweppe, A maximum principle for stochastic control systems, J. Math. Anal. Appl.8 (1964) 287-302.
[52]X. Li. X. Zhou, M. Rami, Indefinite stochastic linear quadratic control with Markovian jumps in infinite time horizon. J. Global Optim.27 (2003) 149-175.
[53]R. Liu, Q. Zhang, G. Yin, Option pricing in a regime-switching model using the fast Fourier transform, J. Appl. Math. Stoch. Anal. (2006) Article ID 18109.22 pages,2006. doi:10.1155/JAMSA/2006/18109.
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