倒向随机微分方程生成元表示定理及其在PDE中的应用
发布时间：2019-05-31 17:20
【摘要】：1990年,Pardoux-Peng[115]提出非线性形式的倒向随机微分方程(Backward Stochastic Differential Equation,简称BSDE),并证明了解的存在唯一性.此后,BSDE理论引起了国内外众多学者的研究兴趣,因为BSDE理论在诸多领域具有重要的应用,如随机分析,偏微分方程(Partial Differential Equation,简称PDE),随机控制,金融数学等.本文主要研究BSDE解的存在唯一性和生成元表示定理,然后利用解的存在唯一性研究连续g-上鞅的非线性Doob-Meyer分解定理,利用生成元表示定理研究二阶非线性PDE的障碍或边值问题,以及状态受限的随机微分对策问题,并介绍一些相关结果.本文的一个主要成果是,PDE粘性解的概率解释问题(Feynman-Kac公式)可以归结为一个BSDE生成元的表示问题.在第2章中,我们首先在生成元g关于y满足弱单调和一般增长条件,关于z满足Lipschitz条件时,采用一种全局截断结合卷积逼近技术证明了一般时间区间(0≤T≤∞)多维BSDE解的存在唯一性;然后利用停时截断区间的方法得到终端时间为无界停时的BSDE解的存在唯一性;接着在同样条件下证明了随机时间区间上一维BSDE解的比较定理;最后,根据这两个随机时间区间的结果,附加一个单边增长条件后,我们证明了一个连续g-上鞅的非线性Doob-Meyer分解定理,由于在T=∞和一般增长条件框架下过程序列缺少弱紧性,导致已有的经典弱收敛技术失效,我们在一个新空间中使用序列的弱相对紧性克服了此困难.在第3章中,我们提出了一个统一的方法—BSDE生成元的表示定理—证明二阶半线性,拟线性及HJB型PDE粘性解的概率解释.我们首先使用表示定理证明了一个二阶拟线性抛物型PDE的Cauchy初值问题粘性解的概率解释,其中漂移系数b(t,x,y,z)=b(t,x,y),扩散系数σ(t,x,y,z)=σ(t,x,y),即二者都不依赖于z.对于b和σ依赖于z的情况,我们将在第4章进行讨论.为了能够体现表示定理方法的优点,我们在生成元的一般增长条件下证明了表示定理,并用此表示定理证明了一个更一般的二阶半线性抛物型PDE的Cauchy初值问题粘性解概率解释.最后,我们用表示定理方法在经典的Lipschitz条件下证明了二阶抛物型HJB方程Cauchy初值问题的粘性解概率解释.通过使用表示定理证明半线性,拟线性和HJB型PDE的Cauchy初值问题粘性解概率解释,我们可将非线性PDE的Cauchy初值问题的粘性解概率解释归结为BSDE生成元的表示问题.在第4章中,我们首先在单调性和Lipschitz条件下使用压缩映射方法证明了带反射的完全耦合正倒向随机微分方程(Forward-Backward Stochastic Differential Equation with Reflections,简写为FBSDER)解的存在唯一性.然后,我们借助于第3章中生成元表示定理方法的思想,使用FBSDER的解证明了二阶拟线性抛物型障碍PDE齐次Neumann边值问题粘性解的存在性.该PDE粘性解的存在性具有以下特点:空间变量的所在区域可以非凸,只需要是有界连通闭集;PDE的二阶导系数可依赖于解的梯度;σ依赖于z时需要求解一个代数方程,求解此代数方程时我们只需要Lipschitz条件,去掉了单调性条件.最后我们在σ(t,x,y,z)=σ(t,x),即σ不依赖于y和z时证明了一个粘性上解和下解的比较原则,由此可得到粘性解的唯一性.在第5章中,我们首先采用时间变换方法证明了一个带有局部时的倒向随机微分方程(称为Generalized BSDE,简称GBSDE)的生成元表示定理.由于GBSDE中存在一个随机测度d Ar,导致经典的表示定理证明过程失效,我们通过时间变换的方法可以将随机测度转化为一个Lebesgue测度dr,从而得到GBSDE与一个鞅驱动的BSDE等价,由此可以将GBSDE生成元的表示问题转换为鞅驱动BSDE的生成元表示问题.然后我们研究了状态受限的两人零和随机微分对策问题,其中状态过程由反射随机微分方程的解给出,状态过程限制在一个有界连通闭集中,代价泛函由GBSDE的解给出.我们得到了值函数的强动态规划原则以及关于初值的正则性.之后,我们使用GBSDE生成元表示定理证明了值函数是一个Isaacs方程非线性Neumann边值问题的粘性解,并通过粘性上解和下解的比较原则得到此粘性解的唯一性.综合第3 5章的内容,我们借助BSDE生成元表示定理这一工具证明了二阶抛物型半线性,拟线性PDE,完全非线性的HJB和Isaacs方程,以及对应的Cauchy初值问题,Neumann边值问题和障碍问题粘性解的概率解释,说明了这几种常见类型的PDE粘性解概率解释问题(非线性Feynman-Kac公式)都可以归结为BSDE生成元表示问题.因此,我们可以称BSDE生成元表示定理方法为求解这些PDE粘性解概率解释的一个统一方法.
[Abstract]:In 1990, Pardoux-Peng[115] proposed a non-linear form of back-to-random differential equation (BSDE) and proved to be unique. Since then, the BSDE theory has aroused the interest of many scholars at home and abroad, because the BSDE theory has important applications in many fields, such as random analysis, partial differential equation (PDE), random control, financial mathematics and so on. In this paper, the existence and uniqueness of the solution of BSDE and the generator's representation theorem are studied. Then, the existence and uniqueness of the solution are used to study the non-linear Doob-Meyer decomposition theorem of the continuous g-upscaling, and the barrier or boundary value problem of the second-order non-linear PDE is studied by using the generator-representation theorem. And introduces some related results. One of the main results of this paper is that the probability interpretation problem of the PDE's viscous solution (the Feynman-Kac formula) can be summed up as a representation of a BSDE generator. In Chapter 2, we first prove the existence and uniqueness of the multi-dimensional BSDE solution of the general time interval (0-T-1) by using a global truncation and convolution approximation technique in the case of the generation of the element g with respect to y to satisfy the weak monotone and general growth conditions, and with respect to z-satisfying the Lipschitz condition. then, the existence and uniqueness of the BSDE solution when the terminal time is the non-boundary stop is obtained by using the method of the time-stop cut-off interval, and then the comparison theorem of one-dimensional BSDE solution on the random time interval is proved under the same condition; and finally, according to the result of the two random time intervals, After a single-sided growth condition is attached, we prove that a continuous g-up-up non-linear Doob-Meyer decomposition theorem results in the failure of the existing classical weak convergence technology due to the lack of weak compactness in the process sequence under the frame of T = 1 and the general growth condition. The weak relative compactness of the use of the sequence in a new space overcomes this difficulty. In Chapter 3, we propose a unified approach to the probability interpretation of the second-order semi-linear, quasi-linear and HJB-type PDE viscous solutions. We first use the representation theorem to demonstrate the probability of a second-order quasi-linear parabolic PDE, with a drift coefficient b (t, x, y, z) = b (t, x, y), a diffusion coefficient of (t, x, y, z) = xt (t, x, y), I. e., both do not rely on z. for both b and z, we will discuss in chapter 4. In order to be able to embody the advantages of the method of the representation theorem, we prove the representation theorem under the general growth condition of the generator, and prove that a more general Cauchy initial viscous solution probability interpretation of the second-order semi-linear parabolic PDE is proved by the theorem. Finally, we use the representation theorem method to prove the viscous solution probability interpretation of the Cauchy initial value of the second order parabolic type HJB equation under the classical Lipschitz condition. By using the representation theorem, we can explain the probability of the initial viscous solution of the Cauchy initial value of the semi-linear, quasi-linear and HJB-type PDE, and we can explain the probability of the initial initial value of the Cauchy initial value of the non-linear PDE as the representation of the BSDE generator. In Chapter 4, we first use the compression mapping method under the condition of monotonicity and Lipschitz to prove the existence and uniqueness of the fully coupled forward-backward stochastic differential equation with reflection (Forward-Backward Stoichitic Differential Equationwith Reflections, abbreviated as FBSDER). Then we use the solution of FBSDER to prove the existence of the viscous solution of the homogeneous Neumann boundary value problem of the second order quasilinear parabolic obstacle. the existence of the pde's viscous solution has the following characteristics: the region of the spatial variable may be non-convex, need only be bounded and closed, the second derivative of the pde can be dependent on the gradient of the solution, We only need the Lipschitz condition when solving this algebraic equation, and the monotonicity condition is removed. In the end, we prove the principle of the comparison of the upper and lower solutions of a viscous solution at the time (t, x, y, z) = xt (t, x), that is, in the absence of y and z, whereby the uniqueness of the viscous solution can be obtained. In Chapter 5, we first use the time transformation method to prove the generator of the inverse stochastic differential equation with local time (called the generalized BSDE, called GBSDE). Due to the existence of a random measure d Ar in the GBSDE, the classical representation theorem proves that the process is invalid, and the random measure can be converted into a Lebesgue measure dr by the method of time transformation, so that the GBSDE is equivalent to a BSDE driven by a driver, Therefore, it is possible to convert the representation problem of the GBSDE generator into a generator for driving the BSDE to represent a problem. Then we study the problem of state-constrained two-person and stochastic differential, in which the state process is given by the solution of the reflection stochastic differential equation, the state process is limited to a bounded communication closed set, and the cost function is given by the solution of GBSDE. We have obtained the strong dynamic programming principle of the value function and the regularity of the initial value. After that, we use the GSDE to generate the meta-representation theorem to prove that the value function is the viscous solution of the non-linear Neumann boundary value problem of the Isaacs equation, and the uniqueness of the viscous solution is obtained by the comparison principle of the viscous upper and lower solutions. Based on the contents of Chapter 3, we use the BSDE to generate the meta-representation theorem. This tool has proved the two-order parabolic semi-linear, quasi-linear PDE, completely non-linear HJB and Isaacs equations, and the corresponding Cauchy problem, Neumann boundary value problem and the probability interpretation of the viscous solution of the obstacle problem. Some common types of PDE viscous solution probability interpretation problems (non-linear Feynman-Kac formula) can be attributed to the BSDE generator representation problem. Therefore, we can call that the BSDE generator is a uniform method for solving the probability of the viscous solution of these PDE.
【学位授予单位】：中国矿业大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O211.63
，
本文编号：2489879
[Abstract]:In 1990, Pardoux-Peng[115] proposed a non-linear form of back-to-random differential equation (BSDE) and proved to be unique. Since then, the BSDE theory has aroused the interest of many scholars at home and abroad, because the BSDE theory has important applications in many fields, such as random analysis, partial differential equation (PDE), random control, financial mathematics and so on. In this paper, the existence and uniqueness of the solution of BSDE and the generator's representation theorem are studied. Then, the existence and uniqueness of the solution are used to study the non-linear Doob-Meyer decomposition theorem of the continuous g-upscaling, and the barrier or boundary value problem of the second-order non-linear PDE is studied by using the generator-representation theorem. And introduces some related results. One of the main results of this paper is that the probability interpretation problem of the PDE's viscous solution (the Feynman-Kac formula) can be summed up as a representation of a BSDE generator. In Chapter 2, we first prove the existence and uniqueness of the multi-dimensional BSDE solution of the general time interval (0-T-1) by using a global truncation and convolution approximation technique in the case of the generation of the element g with respect to y to satisfy the weak monotone and general growth conditions, and with respect to z-satisfying the Lipschitz condition. then, the existence and uniqueness of the BSDE solution when the terminal time is the non-boundary stop is obtained by using the method of the time-stop cut-off interval, and then the comparison theorem of one-dimensional BSDE solution on the random time interval is proved under the same condition; and finally, according to the result of the two random time intervals, After a single-sided growth condition is attached, we prove that a continuous g-up-up non-linear Doob-Meyer decomposition theorem results in the failure of the existing classical weak convergence technology due to the lack of weak compactness in the process sequence under the frame of T = 1 and the general growth condition. The weak relative compactness of the use of the sequence in a new space overcomes this difficulty. In Chapter 3, we propose a unified approach to the probability interpretation of the second-order semi-linear, quasi-linear and HJB-type PDE viscous solutions. We first use the representation theorem to demonstrate the probability of a second-order quasi-linear parabolic PDE, with a drift coefficient b (t, x, y, z) = b (t, x, y), a diffusion coefficient of (t, x, y, z) = xt (t, x, y), I. e., both do not rely on z. for both b and z, we will discuss in chapter 4. In order to be able to embody the advantages of the method of the representation theorem, we prove the representation theorem under the general growth condition of the generator, and prove that a more general Cauchy initial viscous solution probability interpretation of the second-order semi-linear parabolic PDE is proved by the theorem. Finally, we use the representation theorem method to prove the viscous solution probability interpretation of the Cauchy initial value of the second order parabolic type HJB equation under the classical Lipschitz condition. By using the representation theorem, we can explain the probability of the initial viscous solution of the Cauchy initial value of the semi-linear, quasi-linear and HJB-type PDE, and we can explain the probability of the initial initial value of the Cauchy initial value of the non-linear PDE as the representation of the BSDE generator. In Chapter 4, we first use the compression mapping method under the condition of monotonicity and Lipschitz to prove the existence and uniqueness of the fully coupled forward-backward stochastic differential equation with reflection (Forward-Backward Stoichitic Differential Equationwith Reflections, abbreviated as FBSDER). Then we use the solution of FBSDER to prove the existence of the viscous solution of the homogeneous Neumann boundary value problem of the second order quasilinear parabolic obstacle. the existence of the pde's viscous solution has the following characteristics: the region of the spatial variable may be non-convex, need only be bounded and closed, the second derivative of the pde can be dependent on the gradient of the solution, We only need the Lipschitz condition when solving this algebraic equation, and the monotonicity condition is removed. In the end, we prove the principle of the comparison of the upper and lower solutions of a viscous solution at the time (t, x, y, z) = xt (t, x), that is, in the absence of y and z, whereby the uniqueness of the viscous solution can be obtained. In Chapter 5, we first use the time transformation method to prove the generator of the inverse stochastic differential equation with local time (called the generalized BSDE, called GBSDE). Due to the existence of a random measure d Ar in the GBSDE, the classical representation theorem proves that the process is invalid, and the random measure can be converted into a Lebesgue measure dr by the method of time transformation, so that the GBSDE is equivalent to a BSDE driven by a driver, Therefore, it is possible to convert the representation problem of the GBSDE generator into a generator for driving the BSDE to represent a problem. Then we study the problem of state-constrained two-person and stochastic differential, in which the state process is given by the solution of the reflection stochastic differential equation, the state process is limited to a bounded communication closed set, and the cost function is given by the solution of GBSDE. We have obtained the strong dynamic programming principle of the value function and the regularity of the initial value. After that, we use the GSDE to generate the meta-representation theorem to prove that the value function is the viscous solution of the non-linear Neumann boundary value problem of the Isaacs equation, and the uniqueness of the viscous solution is obtained by the comparison principle of the viscous upper and lower solutions. Based on the contents of Chapter 3, we use the BSDE to generate the meta-representation theorem. This tool has proved the two-order parabolic semi-linear, quasi-linear PDE, completely non-linear HJB and Isaacs equations, and the corresponding Cauchy problem, Neumann boundary value problem and the probability interpretation of the viscous solution of the obstacle problem. Some common types of PDE viscous solution probability interpretation problems (non-linear Feynman-Kac formula) can be attributed to the BSDE generator representation problem. Therefore, we can call that the BSDE generator is a uniform method for solving the probability of the viscous solution of these PDE.
【学位授予单位】：中国矿业大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O211.63
，
本文编号：2489879
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