【作者】 赵国庆；

【导师】 陈增敬；

【作者基本信息】 山东大学 ， 金融数学与金融工程， 2010， 博士

【摘要】 通过对期望效应理论的研究,Peng (1997)由非线性的倒向随机微分方程引入了一类非线性期望,g-期望和条件g-期望.近年来,人们越来越认识到了g-期望的强大的应用意义.g-期望的提出不但有着非常重要的非线性数学分析意义也有着广泛的实践应用意义.目前已有大量的文献针对g-期望和g-鞅的不等式进行了研究.比如Chen and Peng (2000)提出了一般g-鞅意义下的下穿不等式,为研究g-鞅意义下的收敛性质提供了一个有力的工具.Chen et al (2003)给出了对所有凸函数成立的g-期望下的Jensen不等式；紧接着Jiang (2005)得出了更一般的g-期望下的Jensen不等式。Wang (2009)研究了g-鞅的最大不等式.我们知道Lenglart控制不等式在半鞅理论和随机分析中发挥着重要的应用。在论文的第一章,我们主要考虑g-期望下的Lenglart控制不等式是否仍然成立或在什么样的条件下g-期望下的Lenglart控制不等式成立?本章对此问题的研究主要动机来源于为求解g-鞅框架下的渐进性质的证明。针对以上问题,在第一章我们将给出肯定的回答。g-期望下的Lenglart控制不等式的结果为我们研究g-鞅(g-期望)框架下的渐进性质提供了一个强有力的工具。通俗来说,最优停时问题就是寻找能够达到最大化期望收益或最小化期望损失的最好停止时刻或决策.最优停时原理在各个领域已有了广泛深入的应用。已有大量的文献针对经典的最优停时原理展开了研究,比如Karatzas and Shreve (1998),Φksendal (1998), Peskir and Shiryaev (2006).然而在某些领域的应用,比如经济金融市场,由于市场中存在大量的模糊事件,Chen and Epstein (2002), Gilboa and Schmeidler (1989)已经指出我们经典的风险为基础的模型在市场中往往是失效的。通过Ellsberg悖论,我们认识到模糊存在的重要性是不能被忽视的,并且在投资决策中,它至少起着与风险同等重要的作用。Riedel (2009)研究了离散时间框架下的多元先验偏好下的最优停时问题,该论文发表在国际顶级期刊Econometrica.受Riedel (2009)启发,在第二章我们发展对应于Riedel (2009)的连续框架下的模糊最优停时原理。该原理利用g-期望来研究我们一般框架下的模糊.我们的目的是在一个充分一般的框架下来研究最优停时,并且我们的框架包含了已有的一些模糊下的连续模型。在连续的框架下我们需要克服更多的技术困难,该框架也有着更广的应用意义.第二章提出的模糊下连续时间最优停时理论为决策者处理各种时间上的最佳策略问题提供了更丰富的洞察力和更强大的工具.在马尔科夫框架下具体求解最优停时问题时,比如美式期权,我们一般是把我们的问题转化为求解一个自由边界问题.在第三章的模糊框架下,我们借助偏微分方程的粘性解来刻画我们的模糊下的值函数和最优停时。对应于模糊下的连续时间最优停时问题,我们得到一类新的自由边界问题.该结果可以被看做Riedal (2009)离散模型中递归方法的推广。在某些无穷时间水平下的应用实例中,我们给出了一些显示解.Royer (2006)引入了一类新的g-期望,该期望是通过由布朗运动和泊松随机测度共同驱动的倒向随机微分方程定义的.在第四章,由Briand et al (2000), Jiang (2005a,2009)工作的启发,我们对应研究新的g-期望和生成元g的性质,并得到新的表示定理和逆比较定理；此外我们还给出了由新的g-期望控制的概率测度集的刻画。以下是本文的结构和主要结论。第一章：我们研究在满足次可加条件下和不满足此条件下的Lenglart控制不等式的具体形式.定理1.2.4(次可加条件下的Lenglart控制不等式)在给定条件(H2),(H3)和(H4)下,已知cadlag适应过程Y在g-期望意义下被一个增过程K L-控制.对(?)∈,δ>0和(?)∈S,那么定理1.3.1(不满足次可加条件下的Lenglart控制不等式)假设cadlag适应过程Y在g-期望意义下被一个增过程K L-控制,其中g只满足条件(H2)-(H3)。对(?)∈,δ>0,(?)∈S,则第二章：在本章我引入模糊下的连续时间框架下的最优停时原理.在第二章中本文主要有三个工作.首先,本文提出一个可行的理论框架.在该框架下本文详细刻画了所求停时问题的值过程,并研究了最优停时问题关于模糊程度的最优律.以下定理是对值过程{Vt}0≤t≤T的一个修正下的刻画：定理2.2.17(Snell包络)在g-期望意义下,{Vt}0≤t≤T是过程{Xt)0≤t≤T的Snell包络.并且过程{Vt}0≤t≤T存在一个RCLL修正{Vt0)0≤t≤T.该过程具有性质V(?)=V(?)0,a.s.,(?)∈S.以下的两个结果是来刻画最优停时.第一个结果是关于g-鞅性质的刻画；而第二个结果是依赖于值过程的描述.定理2.2.18以下的两个命题是等价的：(ⅰ).停时(?)*是最优的,i.e.,εg[X(?)*]=V00=supρ∈Sεg[Xρ]；(ⅱ).存在停时(?)*∈S满足V(?)*0=X(?)*a.s.成立,并且过程{VtΛ(?)*0}0≤t≤T是一个g-鞅.定理2.2.19假设εg[sup0≤t≤T Xt]<∞,且{Kt}0≤t≤T具有连续的路径.则是一个最优停时.其次,当风险厌恶或者模糊厌恶程度改变时,我们可以进行比较我们的停时大小。我们指出越是风险厌恶或越是模糊厌恶的客户越早进行交易.定理2.2.20(模糊最优律1)给定{ct}0≤t≤T,两个生成元g1=g1(c,y,z,w,t),g2=g2(c,y,z,w,t),满足g1≤g2,a.e.,对任意的(y,z,w,t)∈R×Rd×Ω×[0,T].则(?)g1≥(?)g2,a.s.,其中(?)g1,(?)g2分别是对应于g1,g2,的最优停时。定理2.2.21(模糊最优律2)给定生成元序列{gn)n=1∞和g,且满足gn↓g在以下意义下则(?)gn→(?)g,a.s.,其中(?)gn和(?)g分别是关于{gn}n=1∞和g的最优停时.定理2.2.22对于停时问题(2.28),我们有(?)g,U≥(?)g,U2,a.s.,其中(?)g,U和(?)g,U2是对应于U和U2的最优停时.再次,我们给出了具有有界终端时间的与随机终端时间的最优停时值过程之间的关系。定理2.2.23对生成元g满足本文给定的条件下,则有第三章：在马尔科夫的框架下,我们对最优停时问题的值函数进行了更具体的刻画并得到新型的自由边界问题。首先,我们引入一个viscosity smooth-fit条件：命题3.2.3(Viscosity Smooth-fit)对任意的正则点(t,x,c)∈(?),我们有粘性解意义下的表示(?)xu(t,x,c)=(?)xΦ(t,x)具体刻画为．其中次(上)微分的集合定义为作为重要的结果之一,我们引入一个抛物型的自由边界问题来替代解决带确定终端时刻的最优停时问题：定理3.2.4(抛物型的自由边界)假设存在一个连续函数φ=φ(t,x,c)∈[0,T]×Rp×C和一个带正则边界点的开集合(?)={(t,x,c)∈[0,T)×Rp×C：φ(t,x,c)>Φ(t,x)}使得(φ,(?))是如下自由边界问题的一个唯一(粘性)解.其中表示(?)的边界。则并且离开(?)的首逸时(?)是一个最优停时。我们同时也有求解随机终端时刻最优停时问题的椭圆型自由边界问题：定理3.2.5(椭圆型自由边界)假设我们可以找到一个连续函数φ=φ(x,c)∈Rp×C和包含正则点的开集(?)∈Rp×Rl使得(φ,(?))是如下自由边界问题的有界连续(粘性)解：其中g满足和定理2.2.23相同的假设条件。(?)是离开集合(?)的首逸时并且满足P((?)<∞)=1。则停时(?)是最优的.作为本章定理的应用,我们列举两个例子：命题3.3.1(模糊下的最优投资时间)给定所求的价值函数其中ct=(?)St,t>0；(?)<β-b-κσ代表消费对资产的比例。假定b<β+κσ和κ2≤2β.那么最优投资策略的价值函数可以表示为其中此外,最优停时(?)*是项目资产价值达到阈值x0的首中时,其中π1,π2为命题3.3.2(模糊下的最优投入或撤出)对于下列最优停时问题的值函数那么我们可以得到最优投入或撤出策略的值函数为其中x0为x0=λIπ1/π1-1。最优撤出时间为资产价值{St}t≥0达到阈值x0的首中时.参数π1,π2为第四章：本章我们得到一类新的生成元函数9的表示定理,并且给出了由新型g-期望控制的概率集合的一个刻画.我们引入生成元g的一个表示定理：定理4.2.1对每一个(t,x,y,q)∈[0,T[×Rn×R×Rn,生成元g满足假设(R2)-(R5),1≤p≤2,则定理4.2.4假设生成元g是独立于y的并且满足条件(R1),(R3)-(R5),那么我们可以得到S1=S2,其中更多还原

【Abstract】 Motivated by the theory of expected utility, Peng (1997) introduced the notions of g-expectation and conditional g-expectation via a nonlinear backward stochastic differential equation (BSDE). Recently, the interest in g-expectation has remarkably widened, and it has important significance in both nonlinear mathematical theory and many practical applications.There is a substantive literature on the inequalities of g-expectation or g-martingale, see, for example, downcrossing inequality (see Chen and Peng (2000)), Jensen’s inequal-ity (see Chen et al (2003), Jiang (2005)), maximal inequality (see Wang (2009)). It is well known that Lenglart domination inequalities play an important role in the semi-martingale theory and stochastic calculus. In this first chapter, we mainly investigate, under what conditions, Lenglart domination inequalities for g-expectations hold? The original motivation for considering Lenglart domination inequalities for g-expectations comes from the proofs to obtain asymptotic properties in the framework of g-martingale, as well as applications in risk management. In the chapter 1, we shall give an affirmative answer to this question. In the setting of g-martingale (g-expectation), our Lenglart domination inequalities provide a powerful tool to study the asymptotic properties.The optimal stopping time problem is to find the "best" stopping time, or decision, to maximize an expected reward of the gain (or to minimize an expected loss of the loss). As such, the optimal stopping time problem is pervasive in many areas. There is a substantive literature on the theory of classical optimal stopping, see for example Karatzas and Shrcve (1998),Φksendal (1998), Peskir and Shiryaev (2006).However, owing to ambiguity in markets, some kind risk-based models in some lit-erature Chen and Epstein (2002), Gilboa and Schmeidler (1989) have well documented empirical failures in markets. By the Ellsberg Paradox, it is well-known that the im- portance of ambiguity can’t be negligible, and it would be at least as prominent as risk in making investment decisions.Inspired by Riedel (2009) in which a multiprior preference is considered in discrete time, which published in one of international top journals:Econometrica. The purpose of chapter 2 is to develop a counterpart theory of the optimal stopping in continuous time with ambiguity. The technical setting for the theory in continuous time is a g-expectation to represent the ambiguity in a general context. Our purpose is to do so in a context sufficiently general for applications and to encompasses as particular several existing continuous time model under ambiguity. In the continuous time framework, we need to overcome more difficult technical problems, and the framework also has a broader applications. The continuous time theory, presented in chapter 2, offers richer insights and provides more powerful while accessible tools to deal with decision maker’s optimal stoping rule problem.As is well known we can solve an optimal stopping time problem, such as American type option, by transforming to solve a PDE free boundary problem in a Markovian setting. In this setting, we characterize the value function and the optimal stopping rule under ambiguty by using viscosity solution of a partial differential equations (PDE) in chapter 3. We obtain a new class of free boundary problem, which is related to our optimal stopping problem under ambiguity. This result can be viewed as a generaliza-tion of the recursive procedure in discrete time (See Riedal (2009)). In some particular cases, we derive the analytical solutions in the infinite time horizon framework.Royer (2006) introduced a new g-expectation via a nonlinear BSDE with the under-lying filtration generated by both a Brownian motion and a Poisson random measure. In chapter 4, motivated by Briand et al (2000), Jiang (2005a,2009), we studied the properties of the new style of g-expectation and generator g, we obtain a new kind of representation theorem and converse theorem; Furthermore, we give the characteriza-tions of the probability measures dominated by g-expectation.In the following, we list the main result of this thesis.Chapter 1:We mainly introduce the Lenglart domination inequalities for g-expectations both with the sub-additive condition and without this condition.Theorem 1.2.4 (Lenglart domination inequality under sub-additive condi-tion) Under hypotheses(H2), (H3) and (H4), a cadlag adapted process Y is L-dominated by an increasing process K under g-expectation, for (?)ε,δ>0 and (?)∈S, then we haveTheorem 1.3.1 (Lenglart domination inequality without sub-additive condi-tion) Suppose a cadlag adapted process Y is L-dominated by an increasing process K under g-expectation, where g only satisfies (H2)-(H3), for (?)∈,δ>0, (?)∈S, thenChapter 2: In this chapter we develop a theory of optimal stopping time problem in continuous time in the presence of ambiguity.There are three major contributions in this chapter. First, we propose a fairly gen-eral framework. We also present characterization of the value process and the optimal stopping rule for the optimal stopping time problem.The value process {Vt}0≤t≤T is characterized by the following theorem: Theorem 2.2.17 (Snell envelope) {Vt}0≤t＜T is the Snell envelope under g-expectation of the process {Xt}0≤t<T. Furthermore, {Vt}0≤t≤T has a RCLL raodification {Vt0}0≤t≤T with V(?)=V(?)0,a.s., for all (?)∈S.The next two results characterize the optimal stopping rule. The first one charac-terizes the optimal stopping time in terms of g-martingale while the second one depends on the value process. Theorem 2.2.18 The following two statements are equivalent:(ⅰ). A stopping time (?)*, is optimal,(ⅱ). For a stopping time (?)*∈S, we have V(?)*0=X(?)*. a.s. holds, and the process {VtΛ(?)*0}0≤t≤T is a g-martingale.Theorem 2.2.19 Assumeεg[sup0≤t≤T Xt]<∞, and {Xt}0≤t≤T has continuous paths. Then is the optimal stopping time.Second, the theory enables to compare the optimal sopping rule when the risk aversion or the ambiguity changes. We show that while less risk averse agent want to stop late, the agent who is more ambiguous about the market is intend to stop early.Theorem 2.2.20 (Ambiguity and Optimal Rule 1) Given{ct}0≤t≤T, two aggre-gators g1=g1(c,y, z,w,t), g2=g2{c,y,z,w,t), together with g1≤g2, a.e., for all the (y,z,w,t)∈R×Rd×Ω×[0,T]. Then (?)g1≥(?)g2,a.s., where (?)g1, (?)g2 are the optimal stopping times with respect to g1, g2, respectively.Theorem 2.2.21 (Ambiguity and Optimal Rule 2) Given a sequence of aggregators {gn}n=1∞and g with gn↓g in the sense of Then (?)gn→(?)g,a.s., where (?)gn and (?)g are the optimal stopping times with respect to the aggregators{gn}n=1∞and g, respectively.Theorem 2.2.22 For the optimal stopping problem (2.28), we have (?)g,U≥(?)g,U2, a.s., where (?)g,U and (?)g,U2 represents the optimal stopping times of the two agents with respect to U and U2, respectively.Third, we give a relationship between finite time horizon and infinite time horizon.Theorem 2.2.23 Under regularity assumptions on the aggregator, we haveChapter 3:we develop more specifical characterizations of the value function and the optimal stopping rule in a Markovian setting.Firstly, we introduce a viscosity smooth-fit condition:Proposition 3.2.3 (Viscosity Smooth-fit) For all the regular points (t, x, c)∈(?), we have (?)xu(t, x, c)=(?)xΦ(t, x) in the sense of the following inequalities hold: where the set of sub (super) differentials is defined asAs one of mainly results, we introduce a parabolic PDE free boundary problem which solve the problem of optimal stopping with a determinate terminal time horizon:Theorem 3.2.4 (Parabolic free boundary) Given a continuous functionφ=φ(t, x, c)∈[0, T]×Rp×C and an open set (?)={(t,x,c)∈[0, T]×Rp×C:φ(t,x,c)>φ(t,x)} with regular boundary, points such that (φ,(?)) is the unique viscosity solution to the following free boundary, problem: where A:= b(x)(?)x+(σ(x)σT(x))/2(?)x2;(?) is the boundary, of (?). Then and the first exit time (?) from (?) is an optimal stopping time.And PDE free boundary problem which solve the problem of optimal stopping with random time horizon:Theorem 3.2.5 (Elliptic free boundary) Suppose we could find a continuous func-tionφ=φ(x, c)∈Rp×C and an open set (?)∈Rp×Rl such that (φ, (?)) is the bounded continuous viscosity solution to the following free boundary, problem: where g satisfies the same conditions in Theorem 2.2.23. (?) is the first exit time from (?) with P((?)<∞)=1.Then and (?) is an optimal stopping time.As applications of the theorems,we list two cases here:Proposition 3.3.1(Investment timing under ambiguity)Given the value function where ct=(?)St, t>0; (?)<β-b-κσrepresents the proportion of consumption against the asset. Assume b<β+κσandκ2≤2β. Then the value function of the investment strategy is where x0=(σ2I(1-π1)π2)/(σ2(1-π1)(π2-1)-2(?)). Moreover, the optimal stopping time (?)* is the first hitting time of the project value S that hits the threshold x0, whereπ1,π2 are given byProposition 3.3.2(Optimal enter/exit under ambiguity)We know the value function Thus the value function of the optimal enter/exit strategy is where x0 is given by x0=λIπ1/π1-1. The optimal exit time is the first hitting time when {St}t≥0 hits x0. The parametersπ1,π2 areChapter 4: We obtain a new kind of representation theorem for new style of gen-erator g, and characterizations of the probability measures dominated by g-expectation. We claim the following representation theorem:Theorem 4.2.1 For each (t, x, y, q)∈[0, T[;x Rn×R×Rn, let the assumptions (R2)-(R5) hold for the generator g, 1≤p≤2, then we haveTheorem 4.2.4 Suppose the generator g is independent of y and satisfies (R1), (R3)-(R5), thus it follows that S1=S2, where and (?):={(αt,βt)t∈[0,T]:αtz+∫Bβt((?))ut((?))λ(d(?))≤g(t,z,u)}.更多还原