摘要
倒向随机微分方程的历史可追溯到1973年Bismut[4]对随机最优控制问题的研究，而直到Pardoux和Peng在1990年[66]给出了一般形式的倒向随机微分方程以及解的存在唯一性之后，倒向随机微分方程的理论才得到了迅猛的发展，不论在其自身发展方面(参见[20]，[46]，[60]等)还是其应用方面(参见[33]，[34]，[95]等)，倒向随机微分方程都已经成为一个强有力的工具，特别是其在数理金融中的应用，使得倒向随机微分方程越来越受到数学界和金融界的广泛关注(参见[5]，[27]，[51]等)。由[32]，[77]可以看到，期权的定价问题在一定条件下可看作是参数化的倒向随机微分方程的求解问题，即正倒向随机微分方程的求解问题。基于此因，本文第一章首先讨论了一类正倒向随机微分方程解的存在性及其相关问题。期权定价问题是现代金融学研究的重要课题之一，而欧式期权和美式期权是众多期权中最具有代表性的两种。由[32]不难看出，对于时刻T的未定权益ζ，欧式期权的价格是某类倒向随机微分方程零时刻的解。众所周知，在完备的金融市场上未定权益可被自融资交易策略所复制，但是在不完备的金融市场上完全对冲有时很难做到，例如某些衍生证券具有无法完全被对冲的内在风险，这时我们就需要来控制损失，也就是最小化损失发生的概率P(C_T>0)，因而我们将在第二章研究由倒向随机微分方程引导的g-期望和g-概率的一些性质．对应于欧式期权，[65]证明了美式期权的定价实际上就是一个最优停时问题，而在第三章我们将讨论最优停时的相关问题。最后，我们将在第四章通过倒向随机微分方程研究受控的资产组合问题。
总之，本文是围绕倒向随机微分方程和期权定价问题的研究展开的。主要结果具体如下：
1．用迭代的方法证明了无穷区间上带吸收条件的正倒向随机微分方程解的存在唯一性，此结果推广了正倒向随机微分方程可解性的条件。
2．非可加测度的分布不变性、可加性、凸性、次凸性是数量经济和数理金融理论中的重要研究问题。本文讨论了g-期望的分布不变性及g-概率的可加性、凸性、次凸性，得到了一些在金融中有着重要应用的新结果。
3．“high contact原则”问题是经济界研究的一个重点问题之一，本文对由一类倒向随机微分方程引导的最优停时问题和回报函数含初值的最优停时问题进行了研究并得出相应的“high contact原则”。
4．最后我们在Chen[12]工作的基础上，进一步讨论了倒向随机微分方程解Z的性质，给出了关于Z的比较定理并探讨了其在不完备金融市场上的应用。
本文内容包括四章，下面给出每一章的主要内容。
第一章，我们主要研究了如下正倒向随机微分方程适应解的存在唯一性。
其中(x_s，y_s，z_x)取值于R~m×R~1×R~d并且b，σ，c，h是给定的函数。这是一个带吸收条件的无穷区间上的正倒向随机微分方程，而且吸收性系数可以是非光滑的，正向方程的扩散项系数可以是退化的。以往的工作，对于无穷区间上正倒向随机微分方程的研究，单调性条件是一个非常重要的条件例如[78]，而我们这里应用的是吸收条件。并且还举有例子说明有些满足我们所研究条件的方程并不满足单调性条件，这从某种意义上说明，对于该类方程的研究是非常有意义的。下面给出本章的核心结果：
定理1．2．3若假设H1-H3成立，我们考虑正倒向随机微分方程(1．2)，那么(?)u~((n))(t，x)存在并且这个收敛在集合{x∈R~m，0≤t<∞}上是一致的．此外我们还有(x_s~((n),t,x))t≤s<∞在区间[0，∞)依概率1收敛．
这是一迭代定理，最后我们给出上面正倒向随机微分方程的存在性定理和唯—性定理．
定理1．3．2(存在性定理)在引理1．2．2的假设下，方程组(1．1)有解(x(·)，y(·)，z(·))，其中x_s=(?)x_s~((n))，y_s=(?)u~((n+1))(s，x_s~((n)))在区间[t，∞)上依概率1成立；z_s=(?)z_s~((n))且z_s属于空间M~2(0，∞；R~d)．
定理1．3．3(唯一性定理)存在唯一的解u(s，x)和唯一的一对过程(x_s，z_s)使三元组(x_s，u(s，x_s)，z_s)是方程组(1．1)的适应解。
第二章主要研究了三方面的问题。首先讨论了g-期望的分布不变性。在定义g-期望的分布不变性时，有两点值得注意，第一点是由于金融市场可操作性的原因g-期望的分布不变性只针对于马尔科夫过程．另一点是以往定义分布不变性是相对于概率测度P分布相同，而我们研究该问题时分别从概率测度P和g-概率P_g来定义g-期望的分布不变性．我们证明了在相对于概率测度P分布不变时，g-期望具有分布不变性的充分必要条件是g为零．而且分两种情形证明了该结论。
定理2．2．8(独立于y的情形)对于g-期望ε_g[·]，下面的结论是等价的：
(i)ε_g[·]具有分布不变性；
(ii)g(x，t)≡0，(?)(z，t)∈R×[0，T]．
定理2．2．9(一般情形)对于g-期望ε_g[·]，下面的结论等价：
(i)ε_g[·]具有分布不变性；
(ii)g(y，x，t)≡0，(?)(y，z，t)∈R×R×[0，T]．
而在定义相对于g-概率下，我们只得出了g-期望具有分布不变性的充分条件。下面的定理即是我们得出的结果。
定理2．2．11假设存在两个连续函数{α(t)}，{β(t)}使得函数g具有下面的形式：g(y，z，t)=α(t)|z|+β(t)z，
那么g-期望ε_g[·]具有分布不变性。
第二个方面，我们研究了g-概率的可加性。我们知道g-概率是非线性的度量工具，一般来说，如果g不是线性的，g-概率就不是概率测度，那么它也就不具有可加性，但是我们证明了当g=|z|时，g-概率在某些集类上仍保留着概率测度的特征-可加性，这是一个比较有趣的结论。
定理2．3．1对于任意a，b∈R，在倒向随机微分方程(2．33)中，令A={W_T≤a}，B={W_r≤b)，其中r≤T．
(i)z_t~S≤0和z_t~B≤0，a．e．t∈[0，T)；
(ii)z_t~(A∩B)≤0，a．e．t∈[0，T)；
(iii)若k>0，则解y_t~(A∪B)，y_t~A，y_t~B，y_t~(A∩B)在零时刻满足y_0~(A∪B)=y_0~A+y_0~B-y_0~(A∩B)即V(A∪B)+V(A∩B)=V(A)+V(B)．
第三方面我们检验了g-概率与凸容度、次凸容度之间的关系，通过例子，我们可以看到虽然g是凸的，但它并不是凸容度，同时当g满足次可加性时，g-概率是次凸的。这里我们把倒向随机微分方程作为研究g-概率的工具，因此就解决了研究非线性的理论缺乏有力工具的困境。
下面是一个重要的例子，它说明了虽然g是凸的，但由其引导的g-概率并不是凸容度．
例2．4．6令倒向随机微分方程(2．5)中的g=log(1+z~+)和集合A={(W_T-W_r)≤0)，B={W_r≥a}，(?)a∈R，那么V(A∪B)>V(A)+V(B)-V(A∩B)其中V(·)是由倒向随机微分方程引导的g-概率。
在给出主要结果之前，我们先介绍一下次凸的概念。
定义2．4．3[14]如果(?)A，B∈F_T，v(A∪B)≤V(A)+V(B)-V(A∩B)和V(A∪B)≥v(A)+v(B)-v(A∩B)．那么我们称共轭容度V和v是次凸的．
在下面的定理中，我们证明了，当g满足次可加性时，g-概率是次凸的．
定理2．4．7假设倒向随机微分方程的g满足假设(H1)-(H4)，那么由此引导的g-概率是次凸容度
在第三章中我们主要考虑关于最优停时的两个方面的问题。第一个方面我们主要研究了由一类倒向随机微分方程引导的最优停时问题，问题如下所述：
令χ表示所有的停时集合τ≤τ_G．寻找u(x)和τ~*∈χ使得其中g(·)，X_t~x，ε[·]如3．2．1所述，且g在最优停时τ~*达到最大。
对于上面的问题我们研究了值函数的一些性质并给出了对应的“highcontact原则”。
定理3．2．8假设X_t，g及ε(·)如本节开始所述．如果存在一个C~1边界的开集D(?)R~n和一个D上的函数φ(x)，使得：(i)φ(·)∈C~1(D)∩C~2(D)，g(·)∈C~2(G\D)，φ(x)≥g(x) x∈D Ag(x)≤0 x∈(G＼D)；
(ii)(“二阶条件”)(D(x)，g)是下列自由边界问题的解Aφ(x) = 0 x∈Dφ(x)=g(x) x∈(?)D▽_xφ(x)=▽_xg(x) x∈(?)D∩G，则，对于x∈G＼D，通过令φ(x)=g(x)将φ(x)扩展到整个G，我们有其中D={x：φ(x)>g(x)}
第二个方面我们研究了回报函数含有初值的最优停时问题。问题如下：
令χ表示所有的停时集合τ≤τ_G。寻找u(x)和τ~*∈χ使得
其中g(·，x)是R~k中的有界连续可微函数．为了简便，我们仍然假定微分方程(3．2)中的扩散过程X_t满足∑a_(i,j)ζ_iζ_j≥δΣζ_i~2(δ>0)其中a= [a_(i，j)]=1/2σσ~T．
对于该问题我们相应的推导出一个一般的可以用来检验一个给定的函数h是否是(3．6)解的“high contact原则”。我们用两种办法来推导一般“high contact原则”：一个是从实际应用的角度，一个是从直观理论的角度。
定理3．3．5假设X_t，g如本部分开始所述。对任意给定的初始值x，如果存在一个C~1边界的开集D(x)(?)R~k和一个D(x)上的函数q_x(y)，使得：
(i)q_x(·)∈C~1(D(x))∩C~2(D(x))，g(·，x)∈C~2(G＼D(x))，q_x(y)≥g(y，x) y∈D(x) Ag(y，x)≤0 y∈(G\D(x))；
(ii)(“二阶条件”)(D(x)，q)是下列自由边界问题的解Aq_x(y)=0 y∈D(x) q_x(y)=g(y，x) y∈(?)D(x)▽_yq_x=(y)=▽_yg(y，x)y∈(?)D(x)∩G，则，对于y∈G＼D(x)，通过令q_x(y)=g(y，x)将q_x(y)扩展到整个G，我们有其中q(x)=q_x(x)，其中D(x)={y：q_x(y)>g(y，x)}
上面的定理介绍了第一种方法。即把回报函数中所含的初值看作一个参数或常数。下面的定理介绍了第二种方法。即不把它看作常数，而把它看作从x出发的过程。
为此，我们改变Ito过程X_t的初始点从x变为y dX_t=b(X_t)dt+σ(X_t)dB_t，X_0=y，并定义一个常数Ito扩散过程Z_t如下dZ_t=0，Z_0=x．那么，我们得到一个新的R~(2k)中的扩散过程Y_t=Y_t~((y,x))，其中η=(ζ，x)∈R~k×R~k．
于是，Y_t是初始点为(y，x)的扩散过程。另P~((y，x))表示Y_t的概率分布，E~((y,x))表示关于P~((y,x))的期望．由Y_t，方程(3．25)可被写作它是下面问题的一个特殊情形其中u(y，x)的最优停时问题记作τ~*．
因而我们得到下面的定理：
定理3．3．7令g如上所述，X_t同方程(3．37)，G(?)R~k为一开集，记W=G×G．对任意给定的初始点(y，x)，如果存在C~1-边界的开集D(?)R~(2k)和D上的函数φ(y，x)使得：(i)φ(·，x)∈C~1(D)∩C~2(D)，g(·，x)∈C~2(W＼D)，φ(y，x)≥g(y，x) (y，x)∈D Ag(y，x)≤0 (y，x)∈(G＼D)；和(ii)(“二阶条件”)(D，φ)解决下列自由边界问题Aφ(y，x) = 0 (y，x)∈Dφ(y，x)=g(y，x) (y，x)∈(?)D▽_yφ(y，x)=▽_yg(y，x) (y，x)∈(?)D∩W，则通过对(y，x)∈W＼D令φ(y，x)=g(y，x)和令y=x将φ(y，x)扩展到整个W上，我们有其中D={(y，x)：φ(y，x)>g(y，x)}．
在此章的最后，我们还给出了两个例子说明了定理在实际中的应用。
在第四章，我们从[60]中关于推广的Feyman-Kac公式出发，在Chen[12]工作的基础上进一步探讨了Z的性质，在新的条件下得到了关于Z的比较定理，并在此基础上研究了在不完备金融市场上的套期保值问题，进而得到了资产组合可控制区域。下面我们首先给出在更弱的条件下关于Z的表示定理。
定理4．3．1对于正倒向随机微分方程组(4．3)，假设(H3)成立，g∈C_b~(0,1)([0,T]×R~d×R×R~d；R)，Φ在R~d连续．令A={x∈R~d：Φ_x不存在}并进一步假设P{X_T∈A)=0，那么其中(?)，▽X_s是下面变分方程的解．其中I_(n×n)代表n×n单位矩阵．
接着我们得到了Z的比较定理．
定理4．3．3假设(X，Y，Z)是方程组(4．11)的解，(H1-H2)成立．Φ和集合A如定理4．3．1所定义并且σ(s，x)关于x单增，σ(s，x)≥0，(?)(s，x)∈[t,T]×R~n，那么
(i)如果b(s，x)关于x单减，g(s，y，x)分别关于y，z单减并且x(?)A，Φ_x(x)≤K(K≥0)，那么Z_s≤Kσ(s，X_s)a．e．s∈[t，T]
(ii)如果b(s，x)关于x单增，g(s，y，z)分别关于y，z单增，x(?)A，Φ_x(s)≥K(K≥0)，那么Z_s≥Kσ(s，X_s)a．e．s∈[t，T]
The study of Backward Stochastic Differential Equations (BSDEs for short) stems from the research about stochastic control in [4] in 1973. BSDEs hadn't been studied intensely in many directions until the generalized BSDEs were proposed by Pardoux, Peng [66] where the existence and uniqueness result was proved. Since then it has become a powerful tool not only in the fundamental theories (see [20], [46], [60] and so on) but also in applications in finance (see [33], [34], [95] and so on). An important application in mathematical finance is option pricing which received many attentions (see [5], [27], [51] and so on ). From [32], [77], we can see the pricing of options under some conditions can be translated into solving a parameterized BSDE, that is a Forward Backward Stochastic Differential Equation (FBSDE for short). Therefore in Chapter 1 we will consider the theory about FBSDEs. It is well known that European option and American option are the most typical options and in [32], it is easy to see the price of European option can be obtained by solving some kind of BSDE given the European contingent claimζsettled at time T. As we all know, in the complete financial market a given contingent claim can be replicated by a self-financing trading strategy , but in the incomplete financial market the idea of a perfect hedge is limited in scope, i.e. derivative typically will carry an intrinsic risk which can not be hedged away completely. At this stage we are going to focus on controlling the shortfall, that is, minimizing the probability P(C_t > 0) that some shortfall occurs. Then in Chapter 2, we will consider the properties of g-expectation, g-probability inducted by BSDEs. Correspondingly, for American option [65] shows us pricing an American option is in fact an optimal stopping problem which will be explored in Chapter 3. Finally in Chapter 4, we study the constrained portfolios in financial markets by BSDEs.
In a word, the dissertation is developed based on BSDEs and option pricing. Main results are as follows:
1. We prove the existence and uniqueness of an adapted solution of infinite horizon FBSDE with absorption coefficients by successive approximation method, which extends the conditions of solvability of FBSDEs.
2. The law invariance, additivity, 2-alternating and sub 2-alternating of non-additive measure are important research problems in mathematical finance and mathematical economics. And we study the law invariance of g-expectation, the additivity and convexity of g-probability.
3. High contact principle is one of the important problems in economic circle, and we explore the optimal stopping problem inducted by a kind of BSDEs and a class of general optimal stopping problems in which reward functions depend on initial point and derive the corresponding high contact principle.
4. Based on work of Chen [12], the properties of Z in BSDEs and their applications in incomplete financial market are developed.
This dissertation consists of four chapters, whose main contents are described as follows:
In Chapter one we consider the solvability of following forward-backward stochastic differential equations:
where (x_s, y_s, z_s) take values in R~m×R~1×R~d and b,σ, c, h, are given functions with appropriate dimensions; Our aim is to find the unknown {F_t}_(t≥0~-) adapted processes (x(·),y(·),z(·)) which satisfy the above forward-backward stochastic differential equations, on [0,∞), P-almost surety. In this chapter we are interested in this above infinite horizon forward-backward stochastic differential equations. Instead of monotonicity conditions required in the previous work, we consider absorption conditions which were introduced in Preidlin [35]. As an example illustrated, FBSDEs with absorption coefficients are different from those with monotonicity coefficients. By using the successive approximation method, we prove the uniqueness and existence of the solution of FBSDE.
Theorem 1.2.3 Suppose the conditions H1-H3 hold. We consider forward-backward stochastic differential equation (1.2). Then, (?) u~((n))(t, x) exists and this convergence is uniform on the set {x (?) R~m, 0≤t≤∞}. We also have (x_s~((n),t,x))_(t≤s<∞) converges with probability 1 on [0,∞).
This is a successive approximations theorem. Next are the theorems about the uniqueness and existence of the solution of FBSDE.
Theorem 1.3.2 Under the assumptions of Lemma 2.1, equation (1.1) has a solution (x(·),y(·),z(·)) where x_s = (?)x_s~((n)),y_s = (?)u~~n+1))(s,x_s~((n))) with probability 1 on [t,∞); z = (?) z~((n)) in M~2(0,∞; R~d).
Theorem 1.3.3 There exists a unique function u(s,x) and a unique pair of processes (x_s, z_s) such that the triple (x_s, u(s, x_s), z_s) is a solution of equation (1.1).
Chapter two includes three main contents. Firstly, we study the law invariance of g-expectation. When we define the law invariance of g-expectation, there are two points deserving attention. One point is that the law invariance is defined on the set of all random variables with the forms of Markov processes owing to the applied perspective in risk management. The other point is that we define law invariance of g-expectation in the sense thatζ-ηandζ-_gη, respectively. We prove thatε_g [·] is law invariant in the sense thatζ-ηif and only if g is a trivial function. Also the proof is developed in two ways.
Theorem 2.2.8 (The y-independent case) For the g-expexctation, then the following conclusions are equivalent:
(i)ε_g[·] is law invariant;
(ii)g(z,s)≡0,(?)(z,s)∈R×[0,T].
Theorem 2.2.9(The general case) For the g-expexctation, then the following conclusions are equivalent:
(i)ε_g[·] is law invariant;
(ii) g(y,z,s)≡0 ,(?)(y,z,s)∈R×R×[0,T].
But under the definition of the law invariance in the sense thatζ～_gη, we only get the sufficient condition of law invariance of g-expectation.
Theorem 2.2.11 Suppose that there exist two continuous functions {α(t)}, {β(t)} such that g does not depend on y and with the formg(y,z,t) =α(t)|z|+β(t)z, then the g-expectationε_g[·] is law invariant in the sense thatζ～_gη.
Secondly, we study the additivity of g-probability. Capacities are nature and widely used measures of imprecision (or ambiguity) and a generalization of probability measures. Usually capacities fail to have a additivity property except probability measures, so they are not easy to be measured in practice. However, in this section, we consider a natural set of probability measures arising from financial markets. We show that the upper prior probability measures are additive for a large collection of sets. This result is somewhat surprising and unusual.
Theorem 2.3.1 For any a, b∈R, let A = {W_T≤α} and B = {W_τ≥b) for someτ≤T in BSDE , then
(i) z_t~A≤0, and z_t~B≤0, a.e. t∈[0,T).
(ii)z_t~(A∩B)≤0,a.e.t∈[0,T).
(iii) If k > 0, then the solutions y_t~(A∪B), y_t~A, y_t~B, y_t~(A∩B) at time t = 0 satisfyy_0~(A∪B) = y_0~A + y_0~B - y_0~(AB)that is ,V(A∪B) + V(A∩B) = V(A) + V(B).
Thirdly, we study the convexity of g-probability. Most work on capacities has focused on the 2-alternating and sub 2-alternating. It is well known that g-probability defined from peng's g-expectation is a capacity. We show that the g-probability fails to be 2-alternating but is sub 2-alternating when g satisfies the condition of sub-additivity. We have BSDE as the tool to study the properties of the g-probability all through. Thus it is more convenient to study than other capacities.
Next is an important example, this shows although g is convex, the g-probability can not be always 2-alternating.
Example 2.4.6 Let g = log(l+z~+) in BSDE and A = {(W_T- W_τ)≤0}, B = {W_τ≥a}, (?)a∈R. ThenV(A∪B)> V(A) + V(B) - V(A∩B) where the definition of V(·) is defined as the above example.
Before we give the main result, let's introduce the definition of sub 2-alternating.
Definition 2.4.3 We say that conjugate capacities V and v are sub 2-alternating if (?)A, B∈F_T,v(A∪B)≤V(A) + V(B) - V(A∩B)andV(A∪B)≥v(A) + v(B) - v(A∩B).
Theorem 2.4.7 Suppose that g in BSDE satisfies (H1-H4), then the g-probability defined from BSDE is sub 2-alternating.
In Chapter three we consider the problems about optimal stopping. First we study the optimal stopping problem derived by a class of BSDEs. Consider the following problem: find u(x) andτ~*∈χsuch thatwhere g(·) is a bounded continuously differentiate function in R~n.
We study the above question and get the properties of the value function and the high contact principle as follows.
Theorem 3.2.8 Suppose X_t, g andε(·) are described as be described as at the beginning of this section. If there exists an open set D (?) R~n with C~1-boundary and a functionφ(x) on (?) such that: (i)φ(·) G C~1((?))∩C~2(D),g(·)∈C~2(G\(?))φ(x)≥g(x) x∈D Ag(x)≤0 x∈(G\(?));
(ii) ("The second order condition ") (D, q) solves the following free boundaryproblemAφ(x) = 0 x∈Dφ(x)= g(x) x∈(?)D▽_xφ(x) =▽_xg(x) x∈(?)D∩G,Then, extendingφ(x) to all of G by puttingφ(x) = g(x) for x∈G\Dwhere D = {x:φ(x)>g(x)}
Secondly, we consider the stopping time where a reward function g depends not only on X_t, but also on the initial point x. That is, g = g(X_τ~x, x). The qustion is as follows: find u(x) andτ~*∈χsuch sthatwhere g(·,x) is a bounded continuously differentiate function in R~k. For simplicity we also assume that the diffusion process X_t in equation (3.2) satisfiesΣa_(i,j)ζ_iζ_j≥δΣζ_i~2(δ> 0) with a = [a_(i,j)] = 1/2σσ~T.
Next we examine a class of general optimal stopping problems in which reward functions depend on initial points. Two points of view on the initial point are introduced: one is to view it as a constant, the other is to view it as a constant process starting from the point. Based on the two different views, two versions of the generalized high contact principle are derived.
Theorem 3.3.5 Let X_(t, g) be described as at the beginning of this section. For any given initial point x, if there exist an open set D(x) (?) R~k with C~1-boundary and a function q_x(y) on (?)(x) such that: (i) q_x (·)∈C~1((?)(x))∩C~2(D(x)), g(·,x)∈C~2(G\(?)(x)),q_x(y)≥g(y,x) y∈D(x)Ag(y,x)≤0 y∈(G\(?)(x));and (ii) ("The second order condition") (D(x),q) solves the following free boundary problemAq_x(y) = 0 for all y∈D(x)q_x(y) = g(y,x)for all y∈(?)D(x)▽_yq_x(y)=▽_yg(y,x) for all y∈(?)D(x)∩G,then, extending q_x(y) to all of G by putting q_x(y) = g(y, x) for y∈G\D(x), we havewhere q(x) = q_x(x),andD(x) = {y : q_x(y) >g(y,x)}.
So far we have introduced our first point of view on the original initial point x. That is to view it as a parameter or a constant. Next, we will introduce our second point of view on it. That is, instead of viewing it as a constant, to view it as a process starting from x.
To this end, we change the initial point of the Ito diffusion process X_t from x to ydX_t = b(X_t)dt +σ(X_t)dB_t, X_0 = y,and define a constant Ito diffusion process Z_t as followsdZ_t = 0, Z_0 = x. Put them together, we have a new Ito diffusion process Y_t = Y_t~((y,x)) in R~(2k) bywhereη= (ζ,x)∈R~k×R~k. Thus, Y_t is an Ito diffusion process starting at (y,x). Let P~((y,x)) denote the probability law of Y_t and E~((y,x)) denote the expectation w.r.t. P~((y,x)) In terms of Y_t, equation (3.25) can be rewritten as which is a special case of the problemwhere the optimal stopping time associated with u(y,x) is denoted byτ~*.
From the above, we have the second version of the generalized high contact principle:
Theorem 3.3.7 Let g be described as the above and X_t be described as in equation (3.37), and G (?) R~k be an open set and denote W = G×G. For any given initial point (y, z), if there exist an open set D (?) R~(2k) with C~1-boundary and a functionφ(y, x) on (?) such that: (i)φ(·,x)∈C~1((?))∩C~2(D),φ(y,x)≥g(y,x) forall(y,x)∈D Ag(y,x)≤0 for all (y,x)∈(G\(?));and (ii) ("The second order condition") (D,φ) solves the following free boundary problemAφ(y,x)= 0 for all (y, x)∈Dφ(y, x) = g(y, x) for all (y, x)∈(?)D▽_yφ(y,x) =▽_yg(y,x) for all (y,x)∈(?)D∩W,then, extendingφ(y,x) to all of W by puttingφ(y,x) = g(y,x) for (y,x)∈W\D and letting y = x, we havewhereandD = {(y,x) :φ(y,x) > g(y,x)}.
In Chapter 4 we explore the properties of Z in BSDEs. A control theorem with respect to z is obtained. As the application of the results, we get a constrained interval of the portfolio in the incomplete financial market. We first get the new representation theorem.
Theorem 4.3.1 For FBSDE (4.3). suppose (H3) holds, g∈C_b~(0,1)([0, T]×R~d×R×R~d;R),Φis continuous on R~d. Denote A = (x∈R~d :Φ_x dones't exist}. Assume further that P{X_T∈A} = 0. then we havewhere (?),▽X_s is the solution of the variational equationwhere I_(n×n) denotes the nx n identity matrix.
In the following we get a comparison theorem basing on the representation theorem.
Theorem 4.3.3 Suppose (X,Y,Z) is the solution of FBSDE (4.11). assumptions (H1 - H2) hold.Φand the set A are defined as in Theorem 4.3.1 and alsoσ(s, x) is increasing in x andσ(s, x)≥0, (?) (s, x)∈[t, T]×R~n, then
(i) If b(s,x) is decreasing in x, g(s,y,z) is decreasing in y,z respectively, and x(?)A,Φ_x(x)≤K(K≥0), thenZ_s≤K_σ(s,X_s), a.e, s∈[t,T]
(ii) If b(s,x) is increasing in x, g(s,y,z) is increasing in y,z respectively, and x(?)A,Φ_x(x)≥K(K≥0), thenZ_s>Kσ(s,X_s),a.e,s∈[t,T]
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