高维倒向随机微分方程，正倒向随机微分方程及其应用
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摘要
在过去的二十多年里,倒向随机微分方程理论备受大家的关注。线性的倒向随机微分方程是由Bismut ([6]) 1973年首次引入。1990年,Pardoux和Peng ([37])首先证明了非线性倒向随机微分方程的存在唯一性.随后,倒向随机微分方程被广泛的应用于应用及理论方面的研究,特别是在金融数学领域.这篇论文旨在发展倒向随机微分方程和正倒向随机微分方程理论.
在过去的几十年中,虽然倒向随机微分方程理论得到了长足的进步,但是人们还是更多的研究1维倒向随机微分方程,而高维的倒向随机微分方程很少被人所研究.研究高维倒向随机微分方程的最大困难在于没有更加一般的比较定理.实值倒向随机微分方程的比较定理成为该领域比较经典的几个结果之一。该定理最早是由Peng ([41])给出,随后Pardoux和Peng ([38])以及El Karoui-Peng-Quenez ([16])给出了更加一般的结果.比较定理告诉我们：当我们比较实值倒向随机微分方程的两个解得时候,我们只需要比较其生成元及终端值。1994年,在“拟单调条件下”,Christel和Ralf ([12])证明了有限维及无穷维随机微分方程的比较定理.运用类似的方法,1999年,周([60])得到了在有限时间区间内的多维倒向随机微分方程的比较定理.2006年,运用导向随机生存性,Hu和Peng给出了多维倒向随机微分方程成立的充要条件.2009年,运用类似的“拟单调条件下”,Wu-和Xu([52])给出了多维正倒向随机微分方程的比较定理.在第二章,在没有“拟单调条件”的情形下,我们试着去讨论高维倒向随机微分方程的比较定理.我们还将讨论高维的、关于变量z二次增长的倒向随机微分方程,并将研究其对应的偏微分方程.
总所周知,在许多情形下,一个问题的可解性与某一正倒向形式的随机微分方程的可解性是等价的,然而这一形式的正倒向随机微分方程往往超出了现有的结果.另一方面,大家都知道,标准的Lipschitz条件并不能保证正倒向随机微分方程的可解性.因此,大家越来越清醒的认识到,对于这一问题的理解和研究需要更新的视角和观点,大家更加期盼能有一种统一的方法来解决这一问题.
到目前为止,基本上有三种方法去解正倒向随机微分方程：(i)压缩影像方法.这一方法最初是由意大利人Antonelli [2]提出,后来Pardoux和Tang ([36])对这一方法给出了更加细致的解释,但这一方法只能解释当时间区间T相对小的情形.(ii)四步框架法.这一方法最初是由Ma-Protter-Yong([30])给出.对于Markovian形式的正倒向随机微分方程,这一方法是第一个在任意时间区间上给出解得方法.其缺点是要求其系数需要满足一定的光滑性条件,以使得“部分耦合”的偏微分方程有经典解.(iii)连续性方法Hu-Peng[20]及Peng-Wu[45]所提出.后来Yong[54]发展了该方法.该方法可以去解决任意时间区间、非Markovian形式的正倒向随机微分方程.这一方法要求其系数需满足所谓的“单调条件”.这一条件完全的区别于其他的方法.关于这三种方法更加细致的解释可以参阅(cf.[33]).在这本书中将会解释这三种方法之间是不重合的.
第三章,参照[15]and[59]的方法,我们将会对一般的正倒向随机微分方程给出系统的分析.我们的主要工具是使得Yt=u(t,xt)成立的"decoupling fleld"我们需要强调u的一致Lipschitz性在证明过程中起了举足轻重的作用.我们将会给出所谓的"decoupling field"存在的充要条件"decoupling field"保证了正倒向随机微分方程的可解性.我们发现现有的所有的结果,都可以运用我们的结果给出解释,在线性正倒向随机微分方程、系数为确定的情形下,我们的条件也是必须的.
最优控制问题在实际问题当中拥有广泛的应用.在正想控制系统中,布朗运动作为噪声源的线性二次问题是非常著名的控制问题,关于这一问题的经典理论业已得到.最近,倒向线性二次最优控制问题越来越受到大家的关注,比如(Lim,zhou([1]))另一方面,这一问题在经济投资问题当中很容易被提出.例如,在金融市场中,当我们考虑用注入和撤出资金来对冲某一未定权益的时候,研究最优消费选择问题,很自然的就引入了倒向的随机控制问题.
这篇论文旨在研究高维倒向随机微分方程,正倒向随机微分方程及其应用.我们将会得到一类高维倒向随机微分方程的比较定理,得到高维的、生成元关于变量z二次增长、变量y线性增长的倒向随机微分方程的解得存在唯一性.我们还将研究在一般的非Markovian形式的正倒向随机微分方程的可解性.我们得到,现有的大部分的结果都可以用我们的结果给出解释.
本文共分五章,以下是本文的结构和得到的主要结论：
第一章：介绍从第二章到第五章我们讨论的问题,背景,及想法.
Chapter 2:我们将研究一类高维倒向随机微分方程的比较定理,得到高维的、生成元关于变量z二次增长、变量y线性增长的倒向随机微分方程的解得存在唯一性定理2.3.4. (高维倒向随机微分方程的比较定理)设f满足假设2.3.对于任意丁∈0,T],ξ1,ξ2，∈L2(Ω,f,p),(Y1,Z1)和(Y2,Z2)属于Lad2(Ω,C([0,(?)],(?)2))×Lad2(Ω,C([0,(?)],(?)2×d))是BSDE (2.2.3)在终端ξ1和ξ2,在时间[o,τ]上的解,则我们可以找到一线性变换Yt=AtYt使得:当ATξ1≥ATξ2,我们有
定理2.4.3(带线性二次增长的高维倒向随机微分方程的解得存在性)若2.4假定成立.则倒向随机微分方程(2.3.12)有解.
定理2.5.2(带线性二次增长的高维倒向随机微分方程的比较定理)令t(Y,Z)倒向随机微分方程(2.3.12),(Y,Z) BSDE终端值为ξ、生成元为F倒向随机微分方程的解.对于任意的l≥1,倒向随机微分方程的解属于ε×Ml设,对于k=1,2,…,n,P-a.s.,
若Fk满足假定2.4和2.5,则P-a.s.,对于任意的t€[O,T],Ytk≤Ytk.进一步地,如果Y0k=Y0k,则
定理2.5.3(带线性二次增长的高维倒向随机微分方程解的唯一性)若假定2.4and 2.5成立.则倒向随机微分方程(2.3.12)有唯一的解(Ytk,Ztk),并且对于任意的l≥1,Ktk嘱于ε,Ztk属于Ml
定理2.6.4(对应的偏微分方程解得存在性)偏微分方程(2.5.40)的粘性解.
第三章：在这一章中,我们将研究在一般的非Markovian形式的正倒向随机微分方程的可解性.本章的主要目的是建立一种新的框架,将现有的所有的结果统一该框架下.我们的主要工具是所谓的"decoupling random field"而其一致的Lipschitz性将在证明正倒向随机微分方程的可解性的过程中起到举足轻重的作用.通过分析所谓的“特征倒向随机微分方程”,我们可以找到该"decoupling random field"存在的不同条件.我们所研究的“特征倒向随机微分方程”是关于变量z线性二次增长的Riccati方程.我们指出,现有的大部分的结果都可以被我们的结果给出解释.同时,我们还证明了"decoupling random field"的比较性结果.
定义3.2.2若假定3.1成立.我们称u：[0,T]×R×Ω→R是正倒向随机微分方程的" decoupling field",如果
(i)u(T,x)= g(x);
(ⅱ)对于任意的x∈R,u是F-循序可测,并且关于x是一致Lipschitz连续的.
(ⅲ)存在δ:=δ(K0，K)>0,δ只依赖于Lipschitz常数K及K0,使得对于任意的0=t1 定理3.2.3若假定3.1成立.设正倒向随机微分方程(3.0.1)有一"decoupling field"u,则其有唯一解(X,Y.Z)且(1.2.3)成立.
定理3.6.1若假定3.1及(3.3.14)成立.设(3.3.15),(3.3.17)-(3.3.20),(3.3.22)-(3.3.25)中的一个成立.则正倒向随机微分方程(3.0.1)有"decoupling filed"u满足对所有的
因此,正倒向随机微分方程(3.0.1)有唯一的解(X,Y,Z)∈L2使得
推论3.6.2设T给定.若假定3.1成立并且σ=σ(t,x,y)若(3.3.7)-(3.3.9)之一成立,则正倒向随机微分方程(3.0.1)有"decoupling filed"u满足(3.5.1).因此,FBSDE(3.0.1)是可解的.
推论3.6.3若假定3.1成立.若对于(3.1.15)and(3.1.14)定义的参数,有,或则,对于任意的T,正倒向随机微分方程(3.0.1)是可解的.
注3.6.4
(i)Antonelli[2]的工作是假定,任意的以(3.1.15)和(3.1.14)定义的参数σ3和h满足条件(3.3.15)并且时间T足够小.很显然,在本文中,我们的讨论都是依据与此结果.
(ii)Pardoux-Tang[36]的工作,在本质上假定,除了σ3和h满足条件(3.3.15),以及下面的条件之一:
●弱耦合,即,b2,b3,σ2,σ3足够小,f1,h足够小,
·强单调,即,b1足够的负,f2足够的正.
回忆(3.1.10).对于固定的T,第一给条件表明参数y2和y3足够小,这导致常微分方程(3.1.16)在[0,T]上有解.第二个条件表明y的系数足够的负,这保证了常微分方程(3.1.16)在T时刻之前不会爆破.
(ⅲ)Hu-Peng [20],Peng-Wu [45],Yong [54]的工作假设单调条件成立,比如说β>0.通过一些简单的分析,我们立刻就可以发现并且,令△x=0,我们有则我们可以得到(3.5.2)正倒向随机微分方程是可解的.(ⅳ)[59]的工作很显然是推论3.2的一种特殊情况.
(ⅴ)我们需要指出条件(3.3.15),(3.3.17)-(3.3.20),(3.3.22)-(3.3.25)并不能涵盖[30]和[15]的结果.然而,在这些情形下,我们可以通过偏微分方程来讨论确定的"decoupling function"u是一致Lipschitz连续的.
定理3.7.2(稳定性)设(b0,σ0,f0,g0)和(bn,σn,?n,gn),n≥0一致的满足定理3,12的条件.令(Xo,Yo,Zo),(Xn,Yn,Zn)是对应的初始值为x,xn的解,且u,un是对应的随机域.若xn→x0和(bn,σn,?n,gn)→(bo,σo,?o,go),则
定理3.7.3Lp-估计设(b,σ,f,9)满足定理3.12的条件,且Eε>0,使得c1c3<ε足够小,令(X,Y,Z)是以x为初始值的正倒向随机微分方程(3.0.1)的唯一的解,则(?)p≥2,我们有
Chapter 4:在本章中,我们将研究倒向线性二次最优控制问题及倒向非零和微分对策问题.我们可以清晰的得到对应的最优控制及纳什均衡点.同时我们还讨论一类Riccati方程的可解性.所有的这些结果发展了Lim-zhou([1])和Yu-Ji([58])的结果.
定理4.3.1函数是倒向线性二次最优控制问题(4.2.4-4.2.6)的最优解,其中(xt,pt,qt,kt)是以下正倒向随机微分方程的解：
定理4.5.1函数对策问题(4.4.26-4.4.28)的纳什均衡点,其中(xt1,xt2,pt,qt,kt)是以下正倒向随机微分方程的解:
Chapter 5:在本章,我们将研究在白天和晚上不同情形下的企业投资消费选择问题.在这个模型下,投资者将有三种投资选择：投资企业,银行储蓄,及消费.令LX(t)投资者在是时刻t的总资产,π(t)代表投资者投资到企业的财富比例,则(1-π(t))X(t)是投资到无风险债券的比例,投资者的总资产x(t)满足以下方程及
白天,可以选择比例为π的投资和消费率C1去最大化他的财富,而在晚上他不能改变投资比例只能通过不同的消费率C2来最大化他的财富,在晚上投资比例将跟白天的投资比例相同.所以投资者可以通过选择投资策略π和消费率C1,C2来最大化他的财富.
我们考虑一类特殊情形“Hyperbolic Absolute Risk Aversion (HARA)"模型.为了方便起见,我们只考虑时间长度为T+N的情形,T是白天的时间长度,N是晚上的时间长度.令其中γ和R是常数,γ>0,R∈(0,1).我们试着去得到最优决策π,最优消费率C1,C2以及这种情形下的最优值.
定理5.3.1在以上的所有假定下,最优消费选择问题(2.21),(2.22),(3.1),(3.2)的最优解为其中Pt满足常微分方程(5.2.40)和(5.2.50).
The theory of Backward Stochastic Differential Equations (BSDEs) has been ex-tensively studied for the past two decades. This theory can be traced back to Bis-mut ([6]) who studied the linear case. In 1990, Pardoux-Peng ([37]) proved the well-posedness for nonlinear BSDEs. Since then, BSDEs have been extensively studied and used in many applied and theoretical areas, particularly in mathematical finance. The objective of this thesis is to improve and enrich the theory of BSDEs and FBSDEs.
Although BSDEs have received intensive attention during the past decade, peo-ple have much less knowledge of high dimensional BSDEs compared to 1-dimensional BSDEs. The main difficulty for studying high dimensional BSDEs is that there has not a more general comparison theorem. The comparison theorem for real-valued BS-DEs turns out to be one of the classic results of this theory. It was originally stated by S. Peng ([41]) and then generalized by Pardoux-Peng ([38]) and El Karoui-Peng-Quenez ([16]). It allows to compare the solutions of two real-valued BSDEs whenever we can compare the terminal conditions and the generators. In 1994, under "Quasi-monotonicity" assumptions, Christel and Ralf ([12]) proved the comparison theorems for finite and infinite dimensional stochastic differential equations (SDEs in short). Using the similar idea, Zhou ([60]) obtained a comparison theorem for the multi-dimensional BSDEs in finite time intervals in 1999. In 2006, Hu-Peng ([21]) gived a necessary and sufficient condition under which the comparison theorem holds for multidimensional BSDEs and for matrix-valued BSDEs using the Backward Stochastic Viability Property(BSVP). Recently, in 2009, Wu-Xu ([52]) proved the comparison the-orem for high-dimensional FBSDEs, which is also under similar "Quasi-monotonicity" assumptions. In chapter 2, we will try to discuss the comparison theorem for high-dimensional BSDEs, without " Quasi-monotonicity" assumptions. We will also study the high-dimensional BSDEs with quadratic growth in variable z, and related PDEs.
It has been noted that while in many situations the solvability of the original problems is essentially equivalent to the solvability of certain type of Forward Backward Stochastic Differential Equations (FBSDEs in short), these (mostly non-Markovian) FBSDEs are often beyond the scope of any existing frameworks. On the other hand, it is well known that the standard Lipschitz condition is not sufficient for the wellposedness of coupled FBSDEs. Therefore it is becoming increasingly clear that the theory now calls for new insights and ideas that can lead to a better understanding of the problem and hopefully to a unified solution scheme for the general FBSDEs.
There have been three main methods to solve FBSDE (3.0.1):(i) Method of Contraction Mapping. This method, first used by Antonelli [2] and later detailed by Pardoux-Tang ([36]), works well when the duration T is relatively small; (ii) Four Step Scheme. This was the first method that removed restriction on the time duration for Markovian type FBSDEs, initiated by Ma-Protter-Yong ([30]). The trade-off is the requirement on the regularity of the coefficients so that a "decoupling" quasi-linear PDE has a classical solution; and (iii) Method of Continuation. This was a method that can treat non-Markovian FBSDEs with arbitrary duration, initiated by Hu-Peng [20] and Peng-Wu [45], and later developed by Yong [54]. The main assumption for this method is that the coefficients have to satisfy a set of so-called "monotonicity con-ditions", which is restrictive in a different way. We refer to the book (cf. [33]) for the detailed accounts for all three methods. We should remark that these three methods do not cover each other.
In Chapter 3, we shall provide a systematic analysis for the general FBSDE (3.0.1), following the strategy in [15] and [59]. Our main device is a decoupling field u such that Yt = u(t, Xt) holds. We emphasize that the uniform Lipschitz continuity of u is crucial for our purpose. We shall provide a set of sufficient conditions for the existence of such decoupling field, which ultimately leads to the wellposedness of the original FBSDEs. We notice that all the existing frameworks in the literature could be analyzed by using our criteria, and in the linear case with constant coefficients, our conditions are also necessary.
Optimal control theory has developed many applications in practical problems. For forward stochastic system, linear-quadratic control problem with Brownian motion as the noise source is the most popular optimal control problem and the classical theory has been well established. Recently the backward linear-quadratic stochastic control system has received more and more research attention (See e.g. Lim, Zhou ([1])).On the other hand, this kind of control problem naturally arises from financial investment problems. For instance, in financial market, when we consider the problem of hedging one contingent claim using injection and withdrawn of the fund case, the porfolio choice problem forms one kind of backward stochastic optimal control problem.
This thesis is focused on high dimensional BSDEs, FBSDEs and their applications. We will give the comparison theorem of one kind of high dimensional BSDEs and obtain the existence and uniqueness of the solutions for high dimensional BSDEs whose coefficient is of quadratic growth in variable z and of linear growth in variable y. We also study the wellposedness of FBSDEs in a general non-Markovian framework. We show that all the existing frameworks could be analyzed using our new criteria.
This thesis consists of five chapter. In the following, we list the main results of this thesis.
Chapter 1:We introduce problems studied from Chapter 2 to Chapter 5.
Chapter 2:We study the comparison theorem of one kind of high dimensional BSDEs and obtain the existence and uniqueness of high dimensional BSDEs whose coefficient is of quadratic growth in z and of linear growth in y. Theorem 2.3.4. (Comparison Theorem for high-dimensional BSDEs) As-sume f satisfies the Assumption 2.3. Moreover, for an (?)∈[O, T],ξ1ξ2∈L2(Ω,f,P), (Y1,Z1) and(Y2,Z2) in Lad2(Ω,C([O,(?)],(?)2))×Lad2(Ω,C((O,(?)),(?)2×d)) to the BSDE (2.2.3) with terminalξ1 andξ2 over time interval [O, (?)], then we can find a linear transformation Yt = AtYt such that:
if ATξ1≥ATξ2, we have
Theorem 2.4.3 (Existence for high-dimensional BSDEs with quadratic growth) Assume Assumption 2.4 holds. Then BSDE (2.3.12) has a solution.
Theorem 2.5.2 (Comparison Theorem for high-dimensional BSDEs with quadratic growth) Let (Y,Z) be a solution to (2.3.12) and (Y,Z) be a solution to the BSDE associated to the terminal conditionξand to the generator F belong toεx Ml for each l≥1. We assume that, for k = 1,2,…, n, P-a.s., If Fk verifies Assumption 2.4 and 2.5, then P-a.s., for each t∈[0, T], Ytk≤Ytk. If moreover, Y0k = Y0k, then
Theorem 2.5.3 (Uniqueness for high-dimensional BSDEs with quadratic growth) Let the Assumption 2.4 and 2.5 holds. Then BSDE (2.3.12) has a unique solution (Ytk, Ztk) such that Ytk belongs toεand Ztk belongs to Ml for each l≥1.
Theorem 2.6.4 (Existence for related PDEs) (?)(t,x)∈[0,T]×Rm, u(t,x)= Ytt,x is a viscosity solution of PDE (2.5.40).
Chapter 3:In this chapter we study the wellposedness of the FBSDE in a general non-Markovian framework. The main purpose is to build on all the existing methodology in the literature, and put them into a unified scheme. Our main device is a decoupling random field, and its uniform Lipschitz continuity in the spatial variable is crucial for the wellposedness of the original FBSDE. By analyzing a characteristic BSDE, which is a backward stochastic Reccati equation with quadratic growth in the Z component, we find various conditions under which such decoupling random field exists, which lead ultimately to the solvability of the original FBSDE. We show that all the existing frameworks could be analyzed using our new criteria. As a by product, we prove a comparison result for the decoupling field.
Definition 3.2.2 Let Assumption 3.1 holds. We say u : [0,T]×R×Ω→R is a decoupling field of FBSDE if:
(ⅰ) u(T,x)=g(x);
(ⅱ) u is F-progressively measurable for each x∈R, and is uniformly Lipschitz continuous in x with a Lipschitz constant K> 0;
(ⅲ) There exists a constantδ:=δ(K0,K)> 0, which depends only on the Lips-chitz constants K0 and K, such that for any 0=t1< t2≤T with t2-t1≤δ, and anyη∈L2(Ft1), the FBSDE(3.1.2) with initial valueηand terminal condition u(t2,·) has a unique solution, denoted as (Xt1,t2,η,u, Yt1,t2,η,u, Zt1,t2,η,u), which satisfies (1.2.3) for t∈[t1,t2]. Our first result is:
Theorem 3.2.3 Let Assumption 3.1 hold. Assume FBSDE (3.0.1) has a decoupling field u, then it has a unique solution (X,Y,Z) and (1.2.3) holds on [0,T].
Theorem 3.6.1 Let Assumption 3.1 and (3.3.14) hold. Assume one of (3.3,15), (3.3.17)-(3.3.20), (3.3.22)-(3.3.25) holds. Then FBSDE (3.0.1) has a decoupling filed u satisfying Consequently, FBSDE (3.0.1) admits a unique solution (X,Y,Z)€IL2 such that
Corollary 3.6.2 Let T be given. Assume Assumption 3.1 holds andσ=σ(t,x,y). If one of (3.3.7)-(3.3.9) holds, then FBSDE (3.0.1) has a decoupling filed u satisfying (3.5.1). Consequently, FBSDE (3.0.1) is wellposed.
Corollary 3.6.3 Let Assumption 3.1 hold. If, for arbitrary coefficients defined in (3.1.15) and (3.1.14), or then, for any T, FBSDE (3.0.1) is wellposed.
Remark 3.6.4
(ⅰ)The work Antonelli [2] assumes that arbitraryσ3 and h defined in (3.1.15) and (3.1.14) satisfy condition (3.3.15) and T is small enough. This together with Proposition 3.2 implies condition (3.3.15). Of course, in this paper our arguments rely on this result.
(ⅱ) The work Pardoux-Tang [36] essentially assumes, besidesσ3 and h satisfy condition (3.3.15), one of the following conditions:
●weak coupling, that is, either 62, b3.σ2,σ3 are small or fl,h are small; ●strong monotonicity, that is, b1 is very negative or f2 is very negative. Recall (3.1.10). For fixed T, the first condition implies that the coefficients of y2 and y3 is small enough and thus the ODEs (3.1.16) has desired solutions on [0, T]. The second condition implies that the coefficient of y is very negative, which ensures that the solution to ODEs (3.1.16) will not blow up before T.
(ⅲ) The works Hu-Peng [20], Peng-Wu [45], Yong [54] assume certain monotonic-ity condition, e.g. for some constantβ> 0. By some simple analysis, one sees immediately that Moreover, by setting Ax =0, we see that Then This implies (3.5.2) and thus the FBSDE is wellposed.
(ⅳ) The work [59] is clearly a special case of Corollary 3.2.
(ⅴ) We should note that our conditions (3.3.15), (3.3.17)-(3.3.20), (3.3.22)-(3.3.25) do not cover the results in [30] and [15]. However, in these cases by using the PDE arguments the deterministic decoupling function u is uniformly Lipschitz continuous.
Theorem 3.7.2 (Stability) Assume both (b0,σ0, f0, g0) and (bn,σn，fn, gn), n≥0, satisfy the conditions in Theorem 3.12 uniformly. Let (X0，Y0,Z0), (Xn, Yn, Zn) be the corresponding solutions with initial values x, xn, and u. un be the corresponding random fields. Ifxn→x0 and (bn,σn, fn, gn)→(b0,σ0,f0, g0) in appropriate sense, then
Theorem 3.7.3 Lp-estimates Assume (b,σ,?,g) satisfy the conditions in Theorem 3.12, and (?)ε> 0, such that C1C3<εsmall enough, and Let(X,Y,Z) is the unique solution to the FBSDE (3.0.1) with initial ualue x, then (?)p≥2,we have
Chapter 4:In this chapter,we study the linear-quadratic optimal control and a nonzero-sum differential game of backward stochastic differential equations. The optimal control and Nash equilibrium point are explicitly derived.Also the solvability of a kind Riccati equations is discussed.All these results develop that of Lim,Zhou ([1]) and Yu,Ji([58]).
Theorem 4.3.1 The function u(t)= Rt﹣1BT(t)x(t)=Rt﹣1 BT(t)x(t),t∈[0.T],is￡the unique optimal Control for the BLQ problem (4.2.4-4.2.6),where (.r.t,pt,qt,kt) is the solution of the following FBSDEP:
Theorem 4.5.1 The function (ut1,ut2)=((R1)﹣1(B1)Txt1,(R2)﹣1(B2)Txt2),t∈[O,T],is a Nash equilibrium point for the game problem (4.4.26-4.4.28),where(xt1,xt2,pt,qt,kt) is the solution of the following different dimensional FBSDEP:
Chapter 5:In this paper,we present the model of corporate optimal investment with consideration of the difference between the day time and night time.In the model, the investor has three market activities of his choice: corporate investment,savings in a bank, and consumption. Let X(t) denote the total wealth at time t,π(t) represents the proportion of the wealth invested in the real project, then (1 -π(t))X(t) is the amount invested in non-risky bond, and
In the day time, the investor can choose investment with proportionπand con-sumption rate C1 to maximize his wealth, but at the night time, he cannot change his portfolio and only can choose a different consumption rate C2, the portfolio at the night time stays same as the optimal portfolio in the day time. So the investor wants to maximize the following utility of wealth by choosing his investment strategyπand consumption rate C1and C2.
We consider one kind of special case called Hyperbolic Absolute Risk Aversion (HARA) case. For simplicity, we only consider the case where the entire time interval is T+N, T is the duration of the day time, N is the duration of the night time. For general case, the whole interval is n(T+N), we can get the explicit result by repeating the procedure with the same method.
Let hereγand R are constants, whereγ> 0, R∈(0,1). We try to get explicit optimal decisionπ, consumption rates C1, C2 and the optimal value function in this case.
Theorem 5.3.1 Under all the above assumptions, the optimal strategies to the optimal portfolio choice problem (2.21), (2.22), (3.1), (3.2) for the specific HARA case is given by where Pt satisfies ODEs (5.2.40) and (5.2.50).
在过去的几十年中,虽然倒向随机微分方程理论得到了长足的进步,但是人们还是更多的研究1维倒向随机微分方程,而高维的倒向随机微分方程很少被人所研究.研究高维倒向随机微分方程的最大困难在于没有更加一般的比较定理.实值倒向随机微分方程的比较定理成为该领域比较经典的几个结果之一。该定理最早是由Peng ([41])给出,随后Pardoux和Peng ([38])以及El Karoui-Peng-Quenez ([16])给出了更加一般的结果.比较定理告诉我们：当我们比较实值倒向随机微分方程的两个解得时候,我们只需要比较其生成元及终端值。1994年,在“拟单调条件下”,Christel和Ralf ([12])证明了有限维及无穷维随机微分方程的比较定理.运用类似的方法,1999年,周([60])得到了在有限时间区间内的多维倒向随机微分方程的比较定理.2006年,运用导向随机生存性,Hu和Peng给出了多维倒向随机微分方程成立的充要条件.2009年,运用类似的“拟单调条件下”,Wu-和Xu([52])给出了多维正倒向随机微分方程的比较定理.在第二章,在没有“拟单调条件”的情形下,我们试着去讨论高维倒向随机微分方程的比较定理.我们还将讨论高维的、关于变量z二次增长的倒向随机微分方程,并将研究其对应的偏微分方程.
总所周知,在许多情形下,一个问题的可解性与某一正倒向形式的随机微分方程的可解性是等价的,然而这一形式的正倒向随机微分方程往往超出了现有的结果.另一方面,大家都知道,标准的Lipschitz条件并不能保证正倒向随机微分方程的可解性.因此,大家越来越清醒的认识到,对于这一问题的理解和研究需要更新的视角和观点,大家更加期盼能有一种统一的方法来解决这一问题.
到目前为止,基本上有三种方法去解正倒向随机微分方程：(i)压缩影像方法.这一方法最初是由意大利人Antonelli [2]提出,后来Pardoux和Tang ([36])对这一方法给出了更加细致的解释,但这一方法只能解释当时间区间T相对小的情形.(ii)四步框架法.这一方法最初是由Ma-Protter-Yong([30])给出.对于Markovian形式的正倒向随机微分方程,这一方法是第一个在任意时间区间上给出解得方法.其缺点是要求其系数需要满足一定的光滑性条件,以使得“部分耦合”的偏微分方程有经典解.(iii)连续性方法Hu-Peng[20]及Peng-Wu[45]所提出.后来Yong[54]发展了该方法.该方法可以去解决任意时间区间、非Markovian形式的正倒向随机微分方程.这一方法要求其系数需满足所谓的“单调条件”.这一条件完全的区别于其他的方法.关于这三种方法更加细致的解释可以参阅(cf.[33]).在这本书中将会解释这三种方法之间是不重合的.
第三章,参照[15]and[59]的方法,我们将会对一般的正倒向随机微分方程给出系统的分析.我们的主要工具是使得Yt=u(t,xt)成立的"decoupling fleld"我们需要强调u的一致Lipschitz性在证明过程中起了举足轻重的作用.我们将会给出所谓的"decoupling field"存在的充要条件"decoupling field"保证了正倒向随机微分方程的可解性.我们发现现有的所有的结果,都可以运用我们的结果给出解释,在线性正倒向随机微分方程、系数为确定的情形下,我们的条件也是必须的.
最优控制问题在实际问题当中拥有广泛的应用.在正想控制系统中,布朗运动作为噪声源的线性二次问题是非常著名的控制问题,关于这一问题的经典理论业已得到.最近,倒向线性二次最优控制问题越来越受到大家的关注,比如(Lim,zhou([1]))另一方面,这一问题在经济投资问题当中很容易被提出.例如,在金融市场中,当我们考虑用注入和撤出资金来对冲某一未定权益的时候,研究最优消费选择问题,很自然的就引入了倒向的随机控制问题.
这篇论文旨在研究高维倒向随机微分方程,正倒向随机微分方程及其应用.我们将会得到一类高维倒向随机微分方程的比较定理,得到高维的、生成元关于变量z二次增长、变量y线性增长的倒向随机微分方程的解得存在唯一性.我们还将研究在一般的非Markovian形式的正倒向随机微分方程的可解性.我们得到,现有的大部分的结果都可以用我们的结果给出解释.
本文共分五章,以下是本文的结构和得到的主要结论：
第一章：介绍从第二章到第五章我们讨论的问题,背景,及想法.
Chapter 2:我们将研究一类高维倒向随机微分方程的比较定理,得到高维的、生成元关于变量z二次增长、变量y线性增长的倒向随机微分方程的解得存在唯一性定理2.3.4. (高维倒向随机微分方程的比较定理)设f满足假设2.3.对于任意丁∈0,T],ξ1,ξ2，∈L2(Ω,f,p),(Y1,Z1)和(Y2,Z2)属于Lad2(Ω,C([0,(?)],(?)2))×Lad2(Ω,C([0,(?)],(?)2×d))是BSDE (2.2.3)在终端ξ1和ξ2,在时间[o,τ]上的解,则我们可以找到一线性变换Yt=AtYt使得:当ATξ1≥ATξ2,我们有
定理2.4.3(带线性二次增长的高维倒向随机微分方程的解得存在性)若2.4假定成立.则倒向随机微分方程(2.3.12)有解.
定理2.5.2(带线性二次增长的高维倒向随机微分方程的比较定理)令t(Y,Z)倒向随机微分方程(2.3.12),(Y,Z) BSDE终端值为ξ、生成元为F倒向随机微分方程的解.对于任意的l≥1,倒向随机微分方程的解属于ε×Ml设,对于k=1,2,…,n,P-a.s.,
若Fk满足假定2.4和2.5,则P-a.s.,对于任意的t€[O,T],Ytk≤Ytk.进一步地,如果Y0k=Y0k,则
定理2.5.3(带线性二次增长的高维倒向随机微分方程解的唯一性)若假定2.4and 2.5成立.则倒向随机微分方程(2.3.12)有唯一的解(Ytk,Ztk),并且对于任意的l≥1,Ktk嘱于ε,Ztk属于Ml
定理2.6.4(对应的偏微分方程解得存在性)偏微分方程(2.5.40)的粘性解.
第三章：在这一章中,我们将研究在一般的非Markovian形式的正倒向随机微分方程的可解性.本章的主要目的是建立一种新的框架,将现有的所有的结果统一该框架下.我们的主要工具是所谓的"decoupling random field"而其一致的Lipschitz性将在证明正倒向随机微分方程的可解性的过程中起到举足轻重的作用.通过分析所谓的“特征倒向随机微分方程”,我们可以找到该"decoupling random field"存在的不同条件.我们所研究的“特征倒向随机微分方程”是关于变量z线性二次增长的Riccati方程.我们指出,现有的大部分的结果都可以被我们的结果给出解释.同时,我们还证明了"decoupling random field"的比较性结果.
定义3.2.2若假定3.1成立.我们称u：[0,T]×R×Ω→R是正倒向随机微分方程的" decoupling field",如果
(i)u(T,x)= g(x);
(ⅱ)对于任意的x∈R,u是F-循序可测,并且关于x是一致Lipschitz连续的.
(ⅲ)存在δ:=δ(K0，K)>0,δ只依赖于Lipschitz常数K及K0,使得对于任意的0=t1
定理3.6.1若假定3.1及(3.3.14)成立.设(3.3.15),(3.3.17)-(3.3.20),(3.3.22)-(3.3.25)中的一个成立.则正倒向随机微分方程(3.0.1)有"decoupling filed"u满足对所有的
因此,正倒向随机微分方程(3.0.1)有唯一的解(X,Y,Z)∈L2使得
推论3.6.2设T给定.若假定3.1成立并且σ=σ(t,x,y)若(3.3.7)-(3.3.9)之一成立,则正倒向随机微分方程(3.0.1)有"decoupling filed"u满足(3.5.1).因此,FBSDE(3.0.1)是可解的.
推论3.6.3若假定3.1成立.若对于(3.1.15)and(3.1.14)定义的参数,有,或则,对于任意的T,正倒向随机微分方程(3.0.1)是可解的.
注3.6.4
(i)Antonelli[2]的工作是假定,任意的以(3.1.15)和(3.1.14)定义的参数σ3和h满足条件(3.3.15)并且时间T足够小.很显然,在本文中,我们的讨论都是依据与此结果.
(ii)Pardoux-Tang[36]的工作,在本质上假定,除了σ3和h满足条件(3.3.15),以及下面的条件之一:
●弱耦合,即,b2,b3,σ2,σ3足够小,f1,h足够小,
·强单调,即,b1足够的负,f2足够的正.
回忆(3.1.10).对于固定的T,第一给条件表明参数y2和y3足够小,这导致常微分方程(3.1.16)在[0,T]上有解.第二个条件表明y的系数足够的负,这保证了常微分方程(3.1.16)在T时刻之前不会爆破.
(ⅲ)Hu-Peng [20],Peng-Wu [45],Yong [54]的工作假设单调条件成立,比如说β>0.通过一些简单的分析,我们立刻就可以发现并且,令△x=0,我们有则我们可以得到(3.5.2)正倒向随机微分方程是可解的.(ⅳ)[59]的工作很显然是推论3.2的一种特殊情况.
(ⅴ)我们需要指出条件(3.3.15),(3.3.17)-(3.3.20),(3.3.22)-(3.3.25)并不能涵盖[30]和[15]的结果.然而,在这些情形下,我们可以通过偏微分方程来讨论确定的"decoupling function"u是一致Lipschitz连续的.
定理3.7.2(稳定性)设(b0,σ0,f0,g0)和(bn,σn,?n,gn),n≥0一致的满足定理3,12的条件.令(Xo,Yo,Zo),(Xn,Yn,Zn)是对应的初始值为x,xn的解,且u,un是对应的随机域.若xn→x0和(bn,σn,?n,gn)→(bo,σo,?o,go),则
定理3.7.3Lp-估计设(b,σ,f,9)满足定理3.12的条件,且Eε>0,使得c1c3<ε足够小,令(X,Y,Z)是以x为初始值的正倒向随机微分方程(3.0.1)的唯一的解,则(?)p≥2,我们有
Chapter 4:在本章中,我们将研究倒向线性二次最优控制问题及倒向非零和微分对策问题.我们可以清晰的得到对应的最优控制及纳什均衡点.同时我们还讨论一类Riccati方程的可解性.所有的这些结果发展了Lim-zhou([1])和Yu-Ji([58])的结果.
定理4.3.1函数是倒向线性二次最优控制问题(4.2.4-4.2.6)的最优解,其中(xt,pt,qt,kt)是以下正倒向随机微分方程的解：
定理4.5.1函数对策问题(4.4.26-4.4.28)的纳什均衡点,其中(xt1,xt2,pt,qt,kt)是以下正倒向随机微分方程的解:
Chapter 5:在本章,我们将研究在白天和晚上不同情形下的企业投资消费选择问题.在这个模型下,投资者将有三种投资选择：投资企业,银行储蓄,及消费.令LX(t)投资者在是时刻t的总资产,π(t)代表投资者投资到企业的财富比例,则(1-π(t))X(t)是投资到无风险债券的比例,投资者的总资产x(t)满足以下方程及
白天,可以选择比例为π的投资和消费率C1去最大化他的财富,而在晚上他不能改变投资比例只能通过不同的消费率C2来最大化他的财富,在晚上投资比例将跟白天的投资比例相同.所以投资者可以通过选择投资策略π和消费率C1,C2来最大化他的财富.
我们考虑一类特殊情形“Hyperbolic Absolute Risk Aversion (HARA)"模型.为了方便起见,我们只考虑时间长度为T+N的情形,T是白天的时间长度,N是晚上的时间长度.令其中γ和R是常数,γ>0,R∈(0,1).我们试着去得到最优决策π,最优消费率C1,C2以及这种情形下的最优值.
定理5.3.1在以上的所有假定下,最优消费选择问题(2.21),(2.22),(3.1),(3.2)的最优解为其中Pt满足常微分方程(5.2.40)和(5.2.50).
The theory of Backward Stochastic Differential Equations (BSDEs) has been ex-tensively studied for the past two decades. This theory can be traced back to Bis-mut ([6]) who studied the linear case. In 1990, Pardoux-Peng ([37]) proved the well-posedness for nonlinear BSDEs. Since then, BSDEs have been extensively studied and used in many applied and theoretical areas, particularly in mathematical finance. The objective of this thesis is to improve and enrich the theory of BSDEs and FBSDEs.
Although BSDEs have received intensive attention during the past decade, peo-ple have much less knowledge of high dimensional BSDEs compared to 1-dimensional BSDEs. The main difficulty for studying high dimensional BSDEs is that there has not a more general comparison theorem. The comparison theorem for real-valued BS-DEs turns out to be one of the classic results of this theory. It was originally stated by S. Peng ([41]) and then generalized by Pardoux-Peng ([38]) and El Karoui-Peng-Quenez ([16]). It allows to compare the solutions of two real-valued BSDEs whenever we can compare the terminal conditions and the generators. In 1994, under "Quasi-monotonicity" assumptions, Christel and Ralf ([12]) proved the comparison theorems for finite and infinite dimensional stochastic differential equations (SDEs in short). Using the similar idea, Zhou ([60]) obtained a comparison theorem for the multi-dimensional BSDEs in finite time intervals in 1999. In 2006, Hu-Peng ([21]) gived a necessary and sufficient condition under which the comparison theorem holds for multidimensional BSDEs and for matrix-valued BSDEs using the Backward Stochastic Viability Property(BSVP). Recently, in 2009, Wu-Xu ([52]) proved the comparison the-orem for high-dimensional FBSDEs, which is also under similar "Quasi-monotonicity" assumptions. In chapter 2, we will try to discuss the comparison theorem for high-dimensional BSDEs, without " Quasi-monotonicity" assumptions. We will also study the high-dimensional BSDEs with quadratic growth in variable z, and related PDEs.
It has been noted that while in many situations the solvability of the original problems is essentially equivalent to the solvability of certain type of Forward Backward Stochastic Differential Equations (FBSDEs in short), these (mostly non-Markovian) FBSDEs are often beyond the scope of any existing frameworks. On the other hand, it is well known that the standard Lipschitz condition is not sufficient for the wellposedness of coupled FBSDEs. Therefore it is becoming increasingly clear that the theory now calls for new insights and ideas that can lead to a better understanding of the problem and hopefully to a unified solution scheme for the general FBSDEs.
There have been three main methods to solve FBSDE (3.0.1):(i) Method of Contraction Mapping. This method, first used by Antonelli [2] and later detailed by Pardoux-Tang ([36]), works well when the duration T is relatively small; (ii) Four Step Scheme. This was the first method that removed restriction on the time duration for Markovian type FBSDEs, initiated by Ma-Protter-Yong ([30]). The trade-off is the requirement on the regularity of the coefficients so that a "decoupling" quasi-linear PDE has a classical solution; and (iii) Method of Continuation. This was a method that can treat non-Markovian FBSDEs with arbitrary duration, initiated by Hu-Peng [20] and Peng-Wu [45], and later developed by Yong [54]. The main assumption for this method is that the coefficients have to satisfy a set of so-called "monotonicity con-ditions", which is restrictive in a different way. We refer to the book (cf. [33]) for the detailed accounts for all three methods. We should remark that these three methods do not cover each other.
In Chapter 3, we shall provide a systematic analysis for the general FBSDE (3.0.1), following the strategy in [15] and [59]. Our main device is a decoupling field u such that Yt = u(t, Xt) holds. We emphasize that the uniform Lipschitz continuity of u is crucial for our purpose. We shall provide a set of sufficient conditions for the existence of such decoupling field, which ultimately leads to the wellposedness of the original FBSDEs. We notice that all the existing frameworks in the literature could be analyzed by using our criteria, and in the linear case with constant coefficients, our conditions are also necessary.
Optimal control theory has developed many applications in practical problems. For forward stochastic system, linear-quadratic control problem with Brownian motion as the noise source is the most popular optimal control problem and the classical theory has been well established. Recently the backward linear-quadratic stochastic control system has received more and more research attention (See e.g. Lim, Zhou ([1])).On the other hand, this kind of control problem naturally arises from financial investment problems. For instance, in financial market, when we consider the problem of hedging one contingent claim using injection and withdrawn of the fund case, the porfolio choice problem forms one kind of backward stochastic optimal control problem.
This thesis is focused on high dimensional BSDEs, FBSDEs and their applications. We will give the comparison theorem of one kind of high dimensional BSDEs and obtain the existence and uniqueness of the solutions for high dimensional BSDEs whose coefficient is of quadratic growth in variable z and of linear growth in variable y. We also study the wellposedness of FBSDEs in a general non-Markovian framework. We show that all the existing frameworks could be analyzed using our new criteria.
This thesis consists of five chapter. In the following, we list the main results of this thesis.
Chapter 1:We introduce problems studied from Chapter 2 to Chapter 5.
Chapter 2:We study the comparison theorem of one kind of high dimensional BSDEs and obtain the existence and uniqueness of high dimensional BSDEs whose coefficient is of quadratic growth in z and of linear growth in y. Theorem 2.3.4. (Comparison Theorem for high-dimensional BSDEs) As-sume f satisfies the Assumption 2.3. Moreover, for an (?)∈[O, T],ξ1ξ2∈L2(Ω,f,P), (Y1,Z1) and(Y2,Z2) in Lad2(Ω,C([O,(?)],(?)2))×Lad2(Ω,C((O,(?)),(?)2×d)) to the BSDE (2.2.3) with terminalξ1 andξ2 over time interval [O, (?)], then we can find a linear transformation Yt = AtYt such that:
if ATξ1≥ATξ2, we have
Theorem 2.4.3 (Existence for high-dimensional BSDEs with quadratic growth) Assume Assumption 2.4 holds. Then BSDE (2.3.12) has a solution.
Theorem 2.5.2 (Comparison Theorem for high-dimensional BSDEs with quadratic growth) Let (Y,Z) be a solution to (2.3.12) and (Y,Z) be a solution to the BSDE associated to the terminal conditionξand to the generator F belong toεx Ml for each l≥1. We assume that, for k = 1,2,…, n, P-a.s., If Fk verifies Assumption 2.4 and 2.5, then P-a.s., for each t∈[0, T], Ytk≤Ytk. If moreover, Y0k = Y0k, then
Theorem 2.5.3 (Uniqueness for high-dimensional BSDEs with quadratic growth) Let the Assumption 2.4 and 2.5 holds. Then BSDE (2.3.12) has a unique solution (Ytk, Ztk) such that Ytk belongs toεand Ztk belongs to Ml for each l≥1.
Theorem 2.6.4 (Existence for related PDEs) (?)(t,x)∈[0,T]×Rm, u(t,x)= Ytt,x is a viscosity solution of PDE (2.5.40).
Chapter 3:In this chapter we study the wellposedness of the FBSDE in a general non-Markovian framework. The main purpose is to build on all the existing methodology in the literature, and put them into a unified scheme. Our main device is a decoupling random field, and its uniform Lipschitz continuity in the spatial variable is crucial for the wellposedness of the original FBSDE. By analyzing a characteristic BSDE, which is a backward stochastic Reccati equation with quadratic growth in the Z component, we find various conditions under which such decoupling random field exists, which lead ultimately to the solvability of the original FBSDE. We show that all the existing frameworks could be analyzed using our new criteria. As a by product, we prove a comparison result for the decoupling field.
Definition 3.2.2 Let Assumption 3.1 holds. We say u : [0,T]×R×Ω→R is a decoupling field of FBSDE if:
(ⅰ) u(T,x)=g(x);
(ⅱ) u is F-progressively measurable for each x∈R, and is uniformly Lipschitz continuous in x with a Lipschitz constant K> 0;
(ⅲ) There exists a constantδ:=δ(K0,K)> 0, which depends only on the Lips-chitz constants K0 and K, such that for any 0=t1< t2≤T with t2-t1≤δ, and anyη∈L2(Ft1), the FBSDE(3.1.2) with initial valueηand terminal condition u(t2,·) has a unique solution, denoted as (Xt1,t2,η,u, Yt1,t2,η,u, Zt1,t2,η,u), which satisfies (1.2.3) for t∈[t1,t2]. Our first result is:
Theorem 3.2.3 Let Assumption 3.1 hold. Assume FBSDE (3.0.1) has a decoupling field u, then it has a unique solution (X,Y,Z) and (1.2.3) holds on [0,T].
Theorem 3.6.1 Let Assumption 3.1 and (3.3.14) hold. Assume one of (3.3,15), (3.3.17)-(3.3.20), (3.3.22)-(3.3.25) holds. Then FBSDE (3.0.1) has a decoupling filed u satisfying Consequently, FBSDE (3.0.1) admits a unique solution (X,Y,Z)€IL2 such that
Corollary 3.6.2 Let T be given. Assume Assumption 3.1 holds andσ=σ(t,x,y). If one of (3.3.7)-(3.3.9) holds, then FBSDE (3.0.1) has a decoupling filed u satisfying (3.5.1). Consequently, FBSDE (3.0.1) is wellposed.
Corollary 3.6.3 Let Assumption 3.1 hold. If, for arbitrary coefficients defined in (3.1.15) and (3.1.14), or then, for any T, FBSDE (3.0.1) is wellposed.
Remark 3.6.4
(ⅰ)The work Antonelli [2] assumes that arbitraryσ3 and h defined in (3.1.15) and (3.1.14) satisfy condition (3.3.15) and T is small enough. This together with Proposition 3.2 implies condition (3.3.15). Of course, in this paper our arguments rely on this result.
(ⅱ) The work Pardoux-Tang [36] essentially assumes, besidesσ3 and h satisfy condition (3.3.15), one of the following conditions:
●weak coupling, that is, either 62, b3.σ2,σ3 are small or fl,h are small; ●strong monotonicity, that is, b1 is very negative or f2 is very negative. Recall (3.1.10). For fixed T, the first condition implies that the coefficients of y2 and y3 is small enough and thus the ODEs (3.1.16) has desired solutions on [0, T]. The second condition implies that the coefficient of y is very negative, which ensures that the solution to ODEs (3.1.16) will not blow up before T.
(ⅲ) The works Hu-Peng [20], Peng-Wu [45], Yong [54] assume certain monotonic-ity condition, e.g. for some constantβ> 0. By some simple analysis, one sees immediately that Moreover, by setting Ax =0, we see that Then This implies (3.5.2) and thus the FBSDE is wellposed.
(ⅳ) The work [59] is clearly a special case of Corollary 3.2.
(ⅴ) We should note that our conditions (3.3.15), (3.3.17)-(3.3.20), (3.3.22)-(3.3.25) do not cover the results in [30] and [15]. However, in these cases by using the PDE arguments the deterministic decoupling function u is uniformly Lipschitz continuous.
Theorem 3.7.2 (Stability) Assume both (b0,σ0, f0, g0) and (bn,σn，fn, gn), n≥0, satisfy the conditions in Theorem 3.12 uniformly. Let (X0，Y0,Z0), (Xn, Yn, Zn) be the corresponding solutions with initial values x, xn, and u. un be the corresponding random fields. Ifxn→x0 and (bn,σn, fn, gn)→(b0,σ0,f0, g0) in appropriate sense, then
Theorem 3.7.3 Lp-estimates Assume (b,σ,?,g) satisfy the conditions in Theorem 3.12, and (?)ε> 0, such that C1C3<εsmall enough, and Let(X,Y,Z) is the unique solution to the FBSDE (3.0.1) with initial ualue x, then (?)p≥2,we have
Chapter 4:In this chapter,we study the linear-quadratic optimal control and a nonzero-sum differential game of backward stochastic differential equations. The optimal control and Nash equilibrium point are explicitly derived.Also the solvability of a kind Riccati equations is discussed.All these results develop that of Lim,Zhou ([1]) and Yu,Ji([58]).
Theorem 4.3.1 The function u(t)= Rt﹣1BT(t)x(t)=Rt﹣1 BT(t)x(t),t∈[0.T],is￡the unique optimal Control for the BLQ problem (4.2.4-4.2.6),where (.r.t,pt,qt,kt) is the solution of the following FBSDEP:
Theorem 4.5.1 The function (ut1,ut2)=((R1)﹣1(B1)Txt1,(R2)﹣1(B2)Txt2),t∈[O,T],is a Nash equilibrium point for the game problem (4.4.26-4.4.28),where(xt1,xt2,pt,qt,kt) is the solution of the following different dimensional FBSDEP:
Chapter 5:In this paper,we present the model of corporate optimal investment with consideration of the difference between the day time and night time.In the model, the investor has three market activities of his choice: corporate investment,savings in a bank, and consumption. Let X(t) denote the total wealth at time t,π(t) represents the proportion of the wealth invested in the real project, then (1 -π(t))X(t) is the amount invested in non-risky bond, and
In the day time, the investor can choose investment with proportionπand con-sumption rate C1 to maximize his wealth, but at the night time, he cannot change his portfolio and only can choose a different consumption rate C2, the portfolio at the night time stays same as the optimal portfolio in the day time. So the investor wants to maximize the following utility of wealth by choosing his investment strategyπand consumption rate C1and C2.
We consider one kind of special case called Hyperbolic Absolute Risk Aversion (HARA) case. For simplicity, we only consider the case where the entire time interval is T+N, T is the duration of the day time, N is the duration of the night time. For general case, the whole interval is n(T+N), we can get the explicit result by repeating the procedure with the same method.
Let hereγand R are constants, whereγ> 0, R∈(0,1). We try to get explicit optimal decisionπ, consumption rates C1, C2 and the optimal value function in this case.
Theorem 5.3.1 Under all the above assumptions, the optimal strategies to the optimal portfolio choice problem (2.21), (2.22), (3.1), (3.2) for the specific HARA case is given by where Pt satisfies ODEs (5.2.40) and (5.2.50).
引文
[1]Andrew E. B. Lim, X. Zhou, Linear-quadratic control of backward stochastic differential equations, SIAM J. Control Optim.,40 (2001),450-474.
[2]Antonelli, F., Backward-forward Stochastic Differential Equations, Ann. Appl. Probab., 3 (1993), no.3,777-793.
[3]T. Basar, G. J. Olsder, Dynamic noncooperative game theory, London, Academic Press, 1982.
[4]M. Bellalah, Z. Wu, A Simple Model of Corporate International Investment under In-complete Information and Taxes, Annals of Operation Research,165 (2008),123-143.
[5]M. Bellalah, Z. Wu, A model for market closure and international portfolio manage-ment within incomplete information, International Journal of Theoretical and Applied Finance,5 (2002),479-495.
[6]Bismut, J. M. (1973), Theorie Probabiliste du Controle des Diffusions, Mem. Amer. Math. Soc.,176 (1973), Providence, Rhode Island.
[7]P. Briand and Y. Hu, BSDE with quadratic growth and unbounded terminal value, Probab. Theory Related Fields,136 (2006), no.4,604-618.
[8]P. Briand and Y. Hu, Quadratic BSDEs with convex generators and unbounded terminal conditions, Probab. Theory Related Fields,141 (2008),543-567.
[9]P. Briand, J.-P. Lepeltier, J. San Martin, One-dimensional BSDE's whose coefficient is monotonic in y and non-lipschitz in z, Bernoulli,13 (2007), no.1,80-91.
[10]S. Chen, X. Li and X. Zhou, Stochastic linear quadratic regulators with indefinite control weight costs, SIAM J. Control Optim.,36 (1998),1685-1972.
[11]J. J. Choi, Diversification, exchange risks and corporate international investment, Jour-nal of International Business Studies,20 (1989),145-155.
[12]Christel G, Ralf M., Comparison theorems for stochastic differential equations in finite and infinite dimensions, Stoch. Proc. Appl.,53 (1994),23-35.
[13]D. Duffie, Dynamic Asset Pricing Theory, Princeton University Press, Pricenton.
[14]D. Duffie and L. Epstein, Stochastic differential utility, Econometrica,60 (1992),353-394.
[15]F. Delarue, On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stochastic process. Appl.,99 (2002), no 2,209-286.
[16]N. El Karoui, S. Peng, M.C. Quenez, Backward stochastic differential equations in fi-nance, Math. Finance,7 (1997),1-71.
[17]S. Hamadene, Nonzero sum linear-quadratic stochastic differential games and backward-forward equations, Stochastic Anal. appl.,17 (1999),117-130.
[18]M. Kobylanski, Backward stochastic differential equations and partial differential equa-tions with quadratic growth, Ann. Probab.,28 (2000), no.2,558-602.
[19]Y. Hu, J. Ma, and J. Yong, On Semi-linear degenerate backward stochastic partial dif-ferential equations, Probab. Theory Related Fields,123 (2002), no.3,381-411.
[20]Y. Hu and S. Peng, Solution of forward-backward stochastic differential equations, Proba. Theory Rel. Fields,103 (1995),273-283.
[21]Y. Hu, and S. Peng, On the comparison theorem for multidimensional BSDEs, C. R. Acad. Sci. Paris, Ser. I,343 (2006),135-140.
[22]Z. Y. Huang, Z. Wu, One Kind of Corporate International Optimal Investment and Consumption Choice Problem, Proceedings of the 27th Chinese Control Conference, 2008,603-606.
[23]I. Karatzas, Optimization problem in the theory of continuous trading, SIAM Journal of Control and Optimization,27 (1987),1221-1259.
[24]N. Kazamaki and T. Sekiguchi, On the transformation of some classes of martingales by a change of law, Tohoku Math. Journ.,31 (1979),261-279.
[25]M. Kobylanski, Backward stochastic differential equations and partial differential equa-tions with quadratic growth, Ann. Probab.,28 (2000), no.2,558-602.
[26]M. Kobylanski, Resultats d'existence et d'unicite pour des equations differentielles stochastiques retrogrades avec des generateurs a croissance quadratique, C. R. Acad. Sci. Paris Ser. I Math.324 (1997), no.1,81-86.
[27]M. Kohlmann and S. Tang, Global adapted solution of one-dimensional backward stochas-tic Riccati equations, with application to the mean-varaince hedging, Stochastic Process. Appl.,97 (2002),255-288.
[28]J.-P. Lepeltier, J. San Martin, Existence forBSDEwith superlinear-quadratic coefficient, Stochastics Stochastics Rep.,63 (1998),227-240.
[29]P. Briand, J.-P. Lepeltier, J. San Martin, One-dimensional BSDE' s whose coefficient is monotonic in y and non-lipschitz in z, Bernoulli,13 (2007), no.1,80-91.
[30]J. Ma, P. Protter and J. Yong, Solving Forward-backward stochastic differential equations explicitly-A four step scheme, Probab. Theory Relat. Fields.,98 (1994),339-359.
[31]J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications, Stoch. Proc.& Appl.,70 (1997),59-84.
[32]J. Ma and J. Yong, On linear, degenerate backward stochastic partial differential equa-tions, Probab. Theory Relat. Fields,113 (1999),135-170.
[33]J. Ma, and J. Yong, Forward-backward stochastic differential equations and their appli-cations, Lecture Notes in Math.,1702 (1999), Springer.
[34]J. Ma, H. Yin and J. Zhang, On non-Markovian FBSDEs and Backward SPDEs, preprint.
[35]R. Merton, An equilibrium market model with incomplete information, Journal of Fi-nance,42 (1987),483-510.
[36]E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasi-linear parabolic PDEs, Probab. Theory Related Fields,114 (1999), no.2,123-150.
[37]E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, System and Control Letters,14(1990), no.1,55-61.
[38]E. Pardoux, S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, in:Stochastic Partial Differential Equations and their Applications, Charlotte, NC,1991, in:Lecture Notes in Control and Inform. Sci.,176 (1992), Springer, Berlin,200-217.
[39]S. Peng, A general stochastic maximum principle for optimal control problems, SI AM J. Control Optim.,28(1990),966-979.
[40]S. Peng, Probabilistic interpretation for systems of quasilinear parobolic partial differ-ential equations, Stochast. Rep.,37(1991),61-74.
[41]S. Peng, A generalized dynamic programming principle and Hamilton-Jacobi-Bellmen equation, Stochastics Stochastics Rep.,38(1992),119-134
[42]S. Peng, Backward stochastic differential equation and application to optimal control, Applied Mathematics and Optimization,27(1993),125-144
[43]S. Peng, Backward stochastic differential equations-stochastic optimization theory and viscosity solution of HJB equations, Topics on Stochastic Analysis, J. Yan, S. Peng, S. Fang, and L. Wu, eds., Science Press, Beijing,1997,85-138(in Chinese).
[44]S. Peng, Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim.,30 (1992),284-304.
[45]S. Peng, Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to the optimal control, SIAM J. Control Optim.,37(3)(1999),825-843.
[46]S. Tang, General Linear quadratic optimal stochastic control problems with random co-efficents: linear stochastic Hamilton systems and backward stochastic riccati equations, SIAM J. Control Optim.,42 (2003), no.1,53-75.
[47]S. Tang, X. Li, Necessary condition for optimal control of stochastic systems with random jumps, SIAM J. Control Optim.,,32(1994), no.5,1447+1475.
[48]Z. Wu, Forward-backward stochastic differential equations, linear quadratic stochastic optimal control and nonzero sum differential games, Journal of Systems Science and Complexity,18 (2005),179-192.
[49]Z. Wu, Forward-backward stochastic differential equations with Brownian motion and poisson process, Acta Mathematicate Applicatae Sinica,15 (1999), no.3,433-443.
[50]Z. Wu, The Comparison Theorem of FBSDE, Statistics and Probability letters,44 (1999),1-6.
[51]Z. Wu, L. Y. Zhang, The corporate optimal portfolio and consumption choice problem in real project with borrowing rate higher than desposit rate, Applied Mathematics and Commutation,175 (2006),1596-1608.
[52]Z. Wu, M. Xu, Comparison theorems for forward backward SDEs, Statistics and Proba. Letters,79 (2009), no.4,426-435.
[53]Z. Wu, Z. Yu, FBSDE with Poisson process and linear quadratic nonzero sum differential games with random jumps, Applied Mathematics and Mechanics,26 (2005),1034-1039.
[54]J. Yong, Finding adapted solutions of forward-backward stochastic differential equations: method of continuation, Probab. Theory Related Fields,107 (1997), no.4,537-572.
[55]J. Yong, Linear Forward-backward stochastic differential equations, Appl. Math. Optim., 39 (1999), no.1,93-119.
[56]J. Yong, Linear Forward-backward stochastic differential equations with random coeffi-cients, Probab. Theory Related Fields,135 (2006), no.1,53-83.
[57]J. M. Yong, X, Y. Zhou, Stochastic controls:Hamiltonian systems and HJB equations, New York:Springer-Verlag,1999.
[58]Z. Yu, S. Ji, Linear-quadratic nonzero-sum differential games of backward stochastic dif-ferential equations, Proceeding of the 27th chinese control conference, Kunming, Yunan, 2008,562-566.
[59]J. Zhang, The wellposedness of FBSDEs, Dicrete and Continuous Dynamical Systems-Series B,6 (2006), no.4,927-940,.
[60]H. Zhou, Comparison theorem for multi-dimensional backward stochastic differential equations and application, Shandong University Master Thesis,1999.
[2]Antonelli, F., Backward-forward Stochastic Differential Equations, Ann. Appl. Probab., 3 (1993), no.3,777-793.
[3]T. Basar, G. J. Olsder, Dynamic noncooperative game theory, London, Academic Press, 1982.
[4]M. Bellalah, Z. Wu, A Simple Model of Corporate International Investment under In-complete Information and Taxes, Annals of Operation Research,165 (2008),123-143.
[5]M. Bellalah, Z. Wu, A model for market closure and international portfolio manage-ment within incomplete information, International Journal of Theoretical and Applied Finance,5 (2002),479-495.
[6]Bismut, J. M. (1973), Theorie Probabiliste du Controle des Diffusions, Mem. Amer. Math. Soc.,176 (1973), Providence, Rhode Island.
[7]P. Briand and Y. Hu, BSDE with quadratic growth and unbounded terminal value, Probab. Theory Related Fields,136 (2006), no.4,604-618.
[8]P. Briand and Y. Hu, Quadratic BSDEs with convex generators and unbounded terminal conditions, Probab. Theory Related Fields,141 (2008),543-567.
[9]P. Briand, J.-P. Lepeltier, J. San Martin, One-dimensional BSDE's whose coefficient is monotonic in y and non-lipschitz in z, Bernoulli,13 (2007), no.1,80-91.
[10]S. Chen, X. Li and X. Zhou, Stochastic linear quadratic regulators with indefinite control weight costs, SIAM J. Control Optim.,36 (1998),1685-1972.
[11]J. J. Choi, Diversification, exchange risks and corporate international investment, Jour-nal of International Business Studies,20 (1989),145-155.
[12]Christel G, Ralf M., Comparison theorems for stochastic differential equations in finite and infinite dimensions, Stoch. Proc. Appl.,53 (1994),23-35.
[13]D. Duffie, Dynamic Asset Pricing Theory, Princeton University Press, Pricenton.
[14]D. Duffie and L. Epstein, Stochastic differential utility, Econometrica,60 (1992),353-394.
[15]F. Delarue, On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stochastic process. Appl.,99 (2002), no 2,209-286.
[16]N. El Karoui, S. Peng, M.C. Quenez, Backward stochastic differential equations in fi-nance, Math. Finance,7 (1997),1-71.
[17]S. Hamadene, Nonzero sum linear-quadratic stochastic differential games and backward-forward equations, Stochastic Anal. appl.,17 (1999),117-130.
[18]M. Kobylanski, Backward stochastic differential equations and partial differential equa-tions with quadratic growth, Ann. Probab.,28 (2000), no.2,558-602.
[19]Y. Hu, J. Ma, and J. Yong, On Semi-linear degenerate backward stochastic partial dif-ferential equations, Probab. Theory Related Fields,123 (2002), no.3,381-411.
[20]Y. Hu and S. Peng, Solution of forward-backward stochastic differential equations, Proba. Theory Rel. Fields,103 (1995),273-283.
[21]Y. Hu, and S. Peng, On the comparison theorem for multidimensional BSDEs, C. R. Acad. Sci. Paris, Ser. I,343 (2006),135-140.
[22]Z. Y. Huang, Z. Wu, One Kind of Corporate International Optimal Investment and Consumption Choice Problem, Proceedings of the 27th Chinese Control Conference, 2008,603-606.
[23]I. Karatzas, Optimization problem in the theory of continuous trading, SIAM Journal of Control and Optimization,27 (1987),1221-1259.
[24]N. Kazamaki and T. Sekiguchi, On the transformation of some classes of martingales by a change of law, Tohoku Math. Journ.,31 (1979),261-279.
[25]M. Kobylanski, Backward stochastic differential equations and partial differential equa-tions with quadratic growth, Ann. Probab.,28 (2000), no.2,558-602.
[26]M. Kobylanski, Resultats d'existence et d'unicite pour des equations differentielles stochastiques retrogrades avec des generateurs a croissance quadratique, C. R. Acad. Sci. Paris Ser. I Math.324 (1997), no.1,81-86.
[27]M. Kohlmann and S. Tang, Global adapted solution of one-dimensional backward stochas-tic Riccati equations, with application to the mean-varaince hedging, Stochastic Process. Appl.,97 (2002),255-288.
[28]J.-P. Lepeltier, J. San Martin, Existence forBSDEwith superlinear-quadratic coefficient, Stochastics Stochastics Rep.,63 (1998),227-240.
[29]P. Briand, J.-P. Lepeltier, J. San Martin, One-dimensional BSDE' s whose coefficient is monotonic in y and non-lipschitz in z, Bernoulli,13 (2007), no.1,80-91.
[30]J. Ma, P. Protter and J. Yong, Solving Forward-backward stochastic differential equations explicitly-A four step scheme, Probab. Theory Relat. Fields.,98 (1994),339-359.
[31]J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications, Stoch. Proc.& Appl.,70 (1997),59-84.
[32]J. Ma and J. Yong, On linear, degenerate backward stochastic partial differential equa-tions, Probab. Theory Relat. Fields,113 (1999),135-170.
[33]J. Ma, and J. Yong, Forward-backward stochastic differential equations and their appli-cations, Lecture Notes in Math.,1702 (1999), Springer.
[34]J. Ma, H. Yin and J. Zhang, On non-Markovian FBSDEs and Backward SPDEs, preprint.
[35]R. Merton, An equilibrium market model with incomplete information, Journal of Fi-nance,42 (1987),483-510.
[36]E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasi-linear parabolic PDEs, Probab. Theory Related Fields,114 (1999), no.2,123-150.
[37]E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, System and Control Letters,14(1990), no.1,55-61.
[38]E. Pardoux, S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, in:Stochastic Partial Differential Equations and their Applications, Charlotte, NC,1991, in:Lecture Notes in Control and Inform. Sci.,176 (1992), Springer, Berlin,200-217.
[39]S. Peng, A general stochastic maximum principle for optimal control problems, SI AM J. Control Optim.,28(1990),966-979.
[40]S. Peng, Probabilistic interpretation for systems of quasilinear parobolic partial differ-ential equations, Stochast. Rep.,37(1991),61-74.
[41]S. Peng, A generalized dynamic programming principle and Hamilton-Jacobi-Bellmen equation, Stochastics Stochastics Rep.,38(1992),119-134
[42]S. Peng, Backward stochastic differential equation and application to optimal control, Applied Mathematics and Optimization,27(1993),125-144
[43]S. Peng, Backward stochastic differential equations-stochastic optimization theory and viscosity solution of HJB equations, Topics on Stochastic Analysis, J. Yan, S. Peng, S. Fang, and L. Wu, eds., Science Press, Beijing,1997,85-138(in Chinese).
[44]S. Peng, Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim.,30 (1992),284-304.
[45]S. Peng, Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to the optimal control, SIAM J. Control Optim.,37(3)(1999),825-843.
[46]S. Tang, General Linear quadratic optimal stochastic control problems with random co-efficents: linear stochastic Hamilton systems and backward stochastic riccati equations, SIAM J. Control Optim.,42 (2003), no.1,53-75.
[47]S. Tang, X. Li, Necessary condition for optimal control of stochastic systems with random jumps, SIAM J. Control Optim.,,32(1994), no.5,1447+1475.
[48]Z. Wu, Forward-backward stochastic differential equations, linear quadratic stochastic optimal control and nonzero sum differential games, Journal of Systems Science and Complexity,18 (2005),179-192.
[49]Z. Wu, Forward-backward stochastic differential equations with Brownian motion and poisson process, Acta Mathematicate Applicatae Sinica,15 (1999), no.3,433-443.
[50]Z. Wu, The Comparison Theorem of FBSDE, Statistics and Probability letters,44 (1999),1-6.
[51]Z. Wu, L. Y. Zhang, The corporate optimal portfolio and consumption choice problem in real project with borrowing rate higher than desposit rate, Applied Mathematics and Commutation,175 (2006),1596-1608.
[52]Z. Wu, M. Xu, Comparison theorems for forward backward SDEs, Statistics and Proba. Letters,79 (2009), no.4,426-435.
[53]Z. Wu, Z. Yu, FBSDE with Poisson process and linear quadratic nonzero sum differential games with random jumps, Applied Mathematics and Mechanics,26 (2005),1034-1039.
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