【作者】 陈立峰；

【导师】 彭实戈； 赵卫东；

【作者基本信息】 山东大学 ， 概率论与数理统计， 2007， 博士

【摘要】 1973年Bismut[Bismut(1973)]研究了线性形式的倒向随机微分方程(简称BSDE)，1990年Pardoux和Peng给出了BSDE的一般形式，在生成函数满足Lipschitz条件下给出了倒向随机微分方程解的存在唯一性定理。Duffie和Epstein[Duffie & Epstein(1992b)]在研究随机微分效用的过程中也独立引入了一类BSDE的特殊情况。1991年Peng[Peng(1991])]通过倒向随机微分方程得到了非线性Feymann-Kac公式，建立了BSDE理论与偏微分方程理论的紧密联系。1997年，N．El Karoui，Peng和Quenez在[El Karoui et al．(1997b)]中通过BSDE获得了推广的Black-Scholes公式，使得BSDE理论逐渐应用于金融理论中，进一步使得BSDE理论产生了更大的活力。经过十多年的发展，BSDE理论在随机控制、偏微分方程、金融数学、控制论及经济学等领域产生了广泛的应用。BSDE在各个领域的应用需要回答的首要问题就是，当给定终端条件和生成函数时如何求解相应的BSDE。然而正如PDE领域一样，目前只有当生成函数为线性函数或其他几类特殊的非线性函数时才可以得到类似于Black-Scholes公式的显式解，对于大多数情况，我们只能借助于数值方法来求解BSDE。在过去的十几年间出现了一系列求解BSDE的数值方法，从求解原理上总的来说可以分为两大类，第一类方法利用Peng的非线性Feymann-Kac公式，将问题转化为求解与BSDE相对应的拟线型抛物型偏微分方程，此类方法中比较有代表性的是Ma，Protter，Yong[Ma et al．(1994)]及Douglas，Ma，Protter，Yong[Douglas，Jr．etal．(1996)]等人提出的“四步法”，Delarue和Menozzi[Delarue & Menozzi(2006)]的正倒向随机算法，以及Milstein与Tretyakov[Milstein & Tretyakov(2006)]的分层方法。第二类方法直接从BSDE本身的特点出发构造数值格式，此类方法中最早的是V．Bally在[V．Bally(1997)]中引入的时间随机离散格式，还有Chevance[Chevance(1997)]提出的一种求解BSDE的动态规划原理，F．Coquet，V．Mackevi(?)ius，J．Mémin[Coquet，Mackevi(?)ius & Mémin(1998)]，Ph．Briand，B．Delyon，J．Mémin[Briand et al．(2001)]，S．Peng，M．Xu[Peng & Xu(2003)]及J．Ma，Ph．Protter，J．S．Martin，S．Torres[Ma et al．(2002)]等人从不同角度提出了求解BSDE的二叉树方法，并证明了其收敛性，最近的工作有Bouchard & Touzi[Bouchard& Touzi(2004)]给出的使用Malliavin分析来求解BSDE的Monte-Carlo方法，Gobet等人[E．Gobet，JP．Lemor & X．Warin(2005)]在解期权定价问题的最小二乘回归方法的基础上发展出的基于回归的Monte-Carlo方法，以及Zhao，Chen，Peng[WD．Zhao，LF．Chen & SG．Peng(2006)]提出的解BSDE的高精度θ格式。Peng(1997)通过倒向随机微分方程引入了g-期望的概念，g-期望一个很明显的优点是可以很容易定义条件期望。Peng在后续的研究中发现了g-期望这样一个作用于泛函的泛函系统的很多重要性质可以由其生成函数g简单刻画，这对应于金融衍生品市场的意义就是，可以使用不同的g刻画市场上不同类型的交易者，基于Coquet，Hu，Mémin，Peng[Coquet et al．(2002)]的结果，Peng(2004b，d，2005a)提出了动态相容估价和g-估价的理论，并证明了一个动态相容估价满足一些合理假设条件时一定是一个g-估价，这意味着，只要验证金融市场上的一个定价机制满足这些条件，这个定价机制的背后一定存在一个g来刻画其特性，由此而来的两个问题就是：如何验证这些条件?怎样找到市场定价机制背后隐藏的g?对于这样一个反问题，Briand，Coquet，Hu，Mémin & Peng(2000)得到了BSDE的逆比较定理，并给出了一个生成函数g的表示定理，这一结果保证了找到的g的唯一性，表明了定价机制g的可计算性。Yang(2006)使用这一生成函数表示定理来计算定价机制g，得到的(?)与被测试的g吻合的较好，他同时也发现实际金融市场中并不存在支持此计算的条件和数据，所以这个方法只能用来测试，而不能用来计算定价机制g。Yang(2006)将期权价格数据当作随机微分方程(SDE)的一条轨道，使用估计SDE的非参统计方法来估计定价机制g和BSDE的解z，他选用非参核回归方法给出了估计公式，并且进行了数值模拟和估计，得到的结果与统计方法用于估计SDE参数时类似，即对于扩散项(此时对应于BSDE的解z)的估计较有效，但对于漂移项(对应于BSDE的生成函数g)的估计则完全淹没在噪声中，无法得到有效的估计。本文使用一种新的思路来考虑这一问题，方法的思想来源于Peng于2003年得到的g-下鞅分解定理，对于给定的期权的价格过程Y_t，首先找到一个g~μ-估价来控制需寻找的实际市场的g-定价机制，此时Y_t就是一个g~μ-下鞅，使用g-下鞅分解的方法可以近似计算生成函数9，本文对这种方法进行了数值模拟以验证其有效性。金融衍生产品从诞生之日起就一直伴随着风险，金融衍生产品史上曾发生过很多著名的风险事件，早期的有“荷兰郁金香事件”，近期的有“巴林银行事件”、“中航油事件”等，当然这些事件中或多或少有一些其它的风险因素，在这里我们只考虑其中的市场风险，即对于相关产品价格的波动准备不足导致的风险暴露。Artzner，Delbaen，Eber & Heath(1997，1999)引入一致风险度量，将币值型风险度量思想纳入了公理化体系。随后F(?)llmer & Schied(2002a，b，c)和Frittelli & Rosazza Gianin(2002，2004)等人提出了凸风险度量的概念，Rosazza Gianin(2003)利用g-期望引入了一类动态风险度量，Jiang(2005b)给出并证明了一个g-期望是一致风险度量或凸风险度量的充分必要条件。一个好的风险度量其实就是一种强定价机制，可以覆盖因金融产品价格的不确定性产生的风险。收益率的不确定性可以纳入g-期望的范畴，对于波动率的不确定性，Peng(2005b，2006a，b)引入了一类新的非线性期望：G-期望，可以用于风险度量。论文全文共分五章，组织和具体安排如下：第一章主要介绍了BSDE，g-期望基本概念与背景知识。第二章介绍和总结了BSDE数值解法方面前人的工作，对几种有代表性的算法的求解步骤进行了详细的解释与说明。第三章提出了一种求解BSDE的高精度θ-格式，详细描述了格式的构造过程与求解步骤，对几种有代表性的BSDE模型给出了大量的数值结果。第四章第一节介绍了金融衍生产品产生与发展的历史及规模与现状，衍生品市场巨大的交易规模表明了此项研究的迫切性及重要意义，第二节给出了BSDE反问题的描述及相关理论结果，第三节概括了前人的一些相关结果，给出了一种计算BSDE的生成函数g的方法：g-下鞅分解法。第五章介绍了金融风险管理的相关概念，研究了风险度量理论的基本概念及发展过程，对目前国际上普遍使用的SPAN保证金计算系统作了说明，给出了G风险度量的定义，研究了G风险度量参数的设定规则，并使用金融市场数据对SPAN系统与G风险度量做了实证比较。更多还原

【Abstract】 The linear form of Backward Stochastic Differential Equation (BSDE) was first introduced by Bismut (1973). Pardoux & Peng (1990) proved the existence and uniqueness theorem of the solution of nonlinear BSDE when generating function satisfied Lipschitz condition. Duffie & Epstein (1992b) also proposed a type of BSDE independently to characterize the stochastic differential utility. 1991 Peng[Peng (1991)] proposed nonlinear Feymann-Kac formula by using BSDE, established a close link between BSDE and PDE. 1997, N.E1 Karoui, Peng and Quenez[El Karoui et al. (1997b)] obtained the extended Black-Scholes formula, from then on, BSDEs theory applied to financial theory gradually. After 10 years of development, BSDE is further studied and applied widely in stochastic control theory, partial differential equations, mathematical finance, control theory, economics and other fields.BSDEs in various fields of application needs to answer the most important question is : How to solve it When given the terminal conditions and the corresponding generating function.But as PDE areas, Only when the generating function for the other linear or nonlinear function of several special categories, We can only get explicit solutions similar to the Black-Scholes formula, but in most cases, We can only rely on numerical methods for solving BSDEs.In the past 10 years, many efforts have been made in the approximate methods for BSDEs. The principle of these methods can be divided into two categories, the first one transformed BSDEs into corresponding quasi-linear parabolic partial differential equations by using nonlinear Feymann-Kac formula developed by Peng, such as "Four-step scheme" developed by Ma, Protter, Yong[Ma et al. (1994)] and Douglas, Ma, Protter, Yong[Douglas, Jr. et al. (1996)], the forward-backward stochastic algorithms proposed by Deiarue and Menozzi[Delarue & Menozzi (2006)], the layer method developed by Milstein and Tretyakov[Milstein & Tretyakov (2006)]. The second category of methods developed directly from the characteristics of BSDEs. The earliest one among these methods is the random time discrete scheme developed by V.Bally[V. Bally (1997)]. Chevance proposed a dynamic programming principle to solve BSDEs. F.Coquet, V.Mackevicius, J.Memin[Coquet, Mackevicius & Memin (1998)], Ph.Briand, B.Delyon, J.Memin[Briand et al. (2001)], S.Peng, M.Xu[Peng & Xu (2003)] and J.Ma, Ph.Protter, J.S.Martin, S.Torres[Ma et al. (2002)] proposed the Binary Tree method for BSDEs from different points of view, and have proven its convergence. Recently, Bouchard and Touzi[Bouchard& Touzi (2004)] presented a kind of Monte-Carlo method for solving BSDEs by using Malliavin calculus. E.Gobet, JP.Lemor & X.Warin (2005)] proposed a regression based Monte-Carlo method for solving BSDEs. Zhao, Chen, Peng[WD.Zhao, LF.Chen & SG.Peng (2006)] developed a kind of high accurateθ-scheme for BSDEs.Peng (1997) introduced the conception of g-expectation via backward stochastic differential equation, g-expectation has a obvious advantages, it is easy to define conditional expectation by Using Peng’s g-expectation. In the following study, Peng found that some good properties of g- expectations can be simply reflected by its generating function g. For the financial derivatives market, it means that we can using different g indicate Different types of participator in the market. Based on the conclusion of Coquet, Hu, Memin , Peng[Coquet et al. (2002)], Peng (2004b,d, 2005a) elaborated the theory of dynamic consistent evaluation and g-evaluation, he also proved that under some reasonable assumptions a dynamic consistent evaluation is a g-evaluation. This means that if a certification of financial market pricing mechanisms to meet these conditions. Behind the pricing mechanism must exist a g to characterize their characteristics. This resulting two issues : how to verify these conditions? How to find the hidden g behind the market pricing mechanism?For such an inverse problem, Briand, Coquet, Hu, Memin & Peng (2000) proposed the inverse comparison theorem for BSDEs and the representation theorem for generating function g. This result ensures the uniqueness of g been found and the computability of the pricing mechanism g. Yang (2006) has used this representation theorem to calculate pricing mechanism g. The resulting g coincide with the test g well. He also found that in actual financial market there is no such kinds of data and conditions to support this calculation. Therefore, this method can only be used to test and not used to calculate pricing mechanism g.Yang (2006) treated the data of option price as a trajectory of SDE. He used non-parametric statistical methods for SDEs to estimate the pricing mechanisms g and z, the Solutions of BSDEs. He chose Kernel-Regression non-parametric method and given the estimate formula. He used simulation to test his method. The results are similar as the results of SDEs case: For diffusion term (corresponding to the solution z of BSDEs ), the estimation is effective, but for drift term(corresponding to the generating function g of BSDEs ) the estimation completely submerged in the noise. The estimation could not be effective.In this paper, there is a new idea to consider this inverse problem. This method comes from the decomposition theorem of g- submartingale proposed by Peng[2003]. For given option price process Y_t, firstly finding a g~μ-evaluation to dominate the g-pricing mechanism of the actual market. Under such circumstances, Y_t is a g~μ-submartingale, g- submartingale decomposition method can be used approximate generating function g. In this thesis, some numerical simulations have been used to verify the effectiveness of our method.Financial derivative products from the day of its birth, has been accompanied by risks. In the history of financial derivative products, there has been a lot of well-known risk events, such as "Dutch tulips incident", "British Barings Bank incident" and "China Aviation Oil’s case" and so on. Of course, in these incident there is more or less some of other risk factors. Here we consider the market risk only, which means the risk exposure caused by the less of preparation to the price fluctuations of the underlying products. Artzner, Delbaen, Eber &; Heath (1997, 1999) introduce the coherent risk measure. They put Currency type risk-Metric into the axiomatic system. Follmer & Schied (2002a,b,c) and Frittelli & Rosazza Gianin (2002, 2004) followed the concept of a convex risk measure. Rosazza use g- expected to bring in a class of dynamic risk measure, Jianglong presented and proved the necessary and sufficient conditions of g-expectation being a coherent risk measure or convex risk measure.In fact, a good risk measure is a strong pricing mechanism, can cover the risk of financial products caused by the uncertainty of the price. The uncertainty of expected return rate can be included in the scope of g-expectations. For the uncertainty of volatility, Peng (2005b, 2006a,b) introduced a new class of nonlinear expectation: G-expectation, which can be used for this kind of risk measurement.Full paper is divided into five chapters. It’s organized as follows:Chapter 1 introduces the background knowledge and basic concepts of BSDEs and g- expectations.Chapter 2 describes and summarizes some previous works for numerical solution of BSDEs. Several representative algorithms for solving BSDEs were explained in detail. Chapter 3 presents a high accuracyθ-scheme for solving BSDEs, and describes the construction of this scheme in detail. For some representative BSDEs models, a large number of numerical results have been made.Chapter 4 ,section 1 describes the origins and development of the financial derivatives, and the scale and status. The enormous scale of the transaction of Derivatives Market shows the urgency and great significance of this kind of research. Section 2 gives a description of the inverse problem for BSDEs and related theoretical results. Section 3 summarizes the results of previous relevant, gives a calculation method of the generating function g : g- submartingale decomposition method.Chapter 5 introduces some related concepts of financial risk management and the basic concept of risk measurement theory and development.SPAN margin system which is widespread use by the large number of international exchange is explained here. We gives the definition of G-risk Measure and the roles for setting parameters of G-risk measure. Finally, an empirical comparison between SPAN system and G-risk measure system is presented by using of financial market data.更多还原