倒向隨機(jī)微分方程生成元表示定理及其在PDE中的應(yīng)用
發(fā)布時(shí)間：2019-05-31 17:20
【摘要】：1990年,Pardoux-Peng[115]提出非線(xiàn)性形式的倒向隨機(jī)微分方程(Backward Stochastic Differential Equation,簡(jiǎn)稱(chēng)BSDE),并證明了解的存在唯一性.此后,BSDE理論引起了國(guó)內(nèi)外眾多學(xué)者的研究興趣,因?yàn)锽SDE理論在諸多領(lǐng)域具有重要的應(yīng)用,如隨機(jī)分析,偏微分方程(Partial Differential Equation,簡(jiǎn)稱(chēng)PDE),隨機(jī)控制,金融數(shù)學(xué)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連續(xù)g-上鞅的非線(xiàn)性Doob-Meyer分解定理,利用生成元表示定理研究二階非線(xiàn)性PDE的障礙或邊值問(wèn)題,以及狀態(tài)受限的隨機(jī)微分對(duì)策問(wèn)題,并介紹一些相關(guān)結(jié)果.本文的一個(gè)主要成果是,PDE粘性解的概率解釋問(wèn)題(Feynman-Kac公式)可以歸結(jié)為一個(gè)BSDE生成元的表示問(wèn)題.在第2章中,我們首先在生成元g關(guān)于y滿(mǎn)足弱單調(diào)和一般增長(zhǎng)條件,關(guān)于z滿(mǎn)足Lipschitz條件時(shí),采用一種全局截?cái)嘟Y(jié)合卷積逼近技術(shù)證明了一般時(shí)間區(qū)間(0≤T≤∞)多維BSDE解的存在唯一性;然后利用停時(shí)截?cái)鄥^(qū)間的方法得到終端時(shí)間為無(wú)界停時(shí)的BSDE解的存在唯一性;接著在同樣條件下證明了隨機(jī)時(shí)間區(qū)間上一維BSDE解的比較定理;最后,根據(jù)這兩個(gè)隨機(jī)時(shí)間區(qū)間的結(jié)果,附加一個(gè)單邊增長(zhǎng)條件后,我們證明了一個(gè)連續(xù)g-上鞅的非線(xiàn)性Doob-Meyer分解定理,由于在T=∞和一般增長(zhǎng)條件框架下過(guò)程序列缺少弱緊性,導(dǎo)致已有的經(jīng)典弱收斂技術(shù)失效,我們?cè)谝粋�(gè)新空間中使用序列的弱相對(duì)緊性克服了此困難.在第3章中,我們提出了一個(gè)統(tǒng)一的方法—BSDE生成元的表示定理—證明二階半線(xiàn)性,擬線(xiàn)性及HJB型PDE粘性解的概率解釋.我們首先使用表示定理證明了一個(gè)二階擬線(xiàn)性?huà)佄镄蚉DE的Cauchy初值問(wèn)題粘性解的概率解釋,其中漂移系數(shù)b(t,x,y,z)=b(t,x,y),擴(kuò)散系數(shù)σ(t,x,y,z)=σ(t,x,y),即二者都不依賴(lài)于z.對(duì)于b和σ依賴(lài)于z的情況,我們將在第4章進(jìn)行討論.為了能夠體現(xiàn)表示定理方法的優(yōu)點(diǎn),我們?cè)谏稍囊话阍鲩L(zhǎng)條件下證明了表示定理,并用此表示定理證明了一個(gè)更一般的二階半線(xiàn)性?huà)佄镄蚉DE的Cauchy初值問(wèn)題粘性解概率解釋.最后,我們用表示定理方法在經(jīng)典的Lipschitz條件下證明了二階拋物型HJB方程Cauchy初值問(wèn)題的粘性解概率解釋.通過(guò)使用表示定理證明半線(xiàn)性,擬線(xiàn)性和HJB型PDE的Cauchy初值問(wèn)題粘性解概率解釋,我們可將非線(xiàn)性PDE的Cauchy初值問(wèn)題的粘性解概率解釋歸結(jié)為BSDE生成元的表示問(wèn)題.在第4章中,我們首先在單調(diào)性和Lipschitz條件下使用壓縮映射方法證明了帶反射的完全耦合正倒向隨機(jī)微分方程(Forward-Backward Stochastic Differential Equation with Reflections,簡(jiǎn)寫(xiě)為FBSDER)解的存在唯一性.然后,我們借助于第3章中生成元表示定理方法的思想,使用FBSDER的解證明了二階擬線(xiàn)性?huà)佄镄驼系KPDE齊次Neumann邊值問(wèn)題粘性解的存在性.該P(yáng)DE粘性解的存在性具有以下特點(diǎn):空間變量的所在區(qū)域可以非凸,只需要是有界連通閉集;PDE的二階導(dǎo)系數(shù)可依賴(lài)于解的梯度;σ依賴(lài)于z時(shí)需要求解一個(gè)代數(shù)方程,求解此代數(shù)方程時(shí)我們只需要Lipschitz條件,去掉了單調(diào)性條件.最后我們?cè)讦?t,x,y,z)=σ(t,x),即σ不依賴(lài)于y和z時(shí)證明了一個(gè)粘性上解和下解的比較原則,由此可得到粘性解的唯一性.在第5章中,我們首先采用時(shí)間變換方法證明了一個(gè)帶有局部時(shí)的倒向隨機(jī)微分方程(稱(chēng)為Generalized BSDE,簡(jiǎn)稱(chēng)GBSDE)的生成元表示定理.由于GBSDE中存在一個(gè)隨機(jī)測(cè)度d Ar,導(dǎo)致經(jīng)典的表示定理證明過(guò)程失效,我們通過(guò)時(shí)間變換的方法可以將隨機(jī)測(cè)度轉(zhuǎn)化為一個(gè)Lebesgue測(cè)度dr,從而得到GBSDE與一個(gè)鞅驅(qū)動(dòng)的BSDE等價(jià),由此可以將GBSDE生成元的表示問(wèn)題轉(zhuǎn)換為鞅驅(qū)動(dòng)BSDE的生成元表示問(wèn)題.然后我們研究了狀態(tài)受限的兩人零和隨機(jī)微分對(duì)策問(wèn)題,其中狀態(tài)過(guò)程由反射隨機(jī)微分方程的解給出,狀態(tài)過(guò)程限制在一個(gè)有界連通閉集中,代價(jià)泛函由GBSDE的解給出.我們得到了值函數(shù)的強(qiáng)動(dòng)態(tài)規(guī)劃原則以及關(guān)于初值的正則性.之后,我們使用GBSDE生成元表示定理證明了值函數(shù)是一個(gè)Isaacs方程非線(xiàn)性Neumann邊值問(wèn)題的粘性解,并通過(guò)粘性上解和下解的比較原則得到此粘性解的唯一性.綜合第3 5章的內(nèi)容,我們借助BSDE生成元表示定理這一工具證明了二階拋物型半線(xiàn)性,擬線(xiàn)性PDE,完全非線(xiàn)性的HJB和Isaacs方程,以及對(duì)應(yīng)的Cauchy初值問(wèn)題,Neumann邊值問(wèn)題和障礙問(wèn)題粘性解的概率解釋,說(shuō)明了這幾種常見(jiàn)類(lèi)型的PDE粘性解概率解釋問(wèn)題(非線(xiàn)性Feynman-Kac公式)都可以歸結(jié)為BSDE生成元表示問(wèn)題.因此,我們可以稱(chēng)BSDE生成元表示定理方法為求解這些PDE粘性解概率解釋的一個(gè)統(tǒng)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.
【學(xué)位授予單位】：中國(guó)礦業(yè)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O211.63
，
本文編號(hào)：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.
【學(xué)位授予單位】：中國(guó)礦業(yè)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O211.63
，
本文編號(hào)：2489879
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