多过程驱动的随机常微分方程几类终值与边值问题适应解性质的研究
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摘要
本学位论文旨在研究多过程驱动的随机常微分方程几类终值与边值问题适应解的性质,其中的驱动过程由三个独立的随机过程组成(两个维纳过程Wt,Bt与一个跳过程Lt),而对应的σ域全体Ft既不单增也不单降,故不构成一般的域流.全文共分三大部分.第一部分,通过证明带跳的一般Skorohod积分意义下的Ito公式和Doleans-Dade公式,首次证明了由Levy过程驱动的倒向双重随机微分方程,当系数函数f与Z有关时的比较定理,我们给出的条件为易于判定的“负跳一致小”条件.第二部分,证明了由Levy过程驱动的倒向随机微分方程在不同Bihari条件下适应解的存在唯一性.第三部分,不同于Pardoux和Peng的鞅表示定理,本文通过扩大解容许范围引入变差逼近的思想,研究随机微分方程两点边值问题适应解的存在性.给出方程适应解存在的充要条件,同时利用构造解序列的方法给出方程的一个适应解.为此全文共分五章.
第一章给出本学位论文所需要的预备知识,包括Levy过程,Skorohod积分以及Malliavin微分的一些主要结论和性质.
第二章讨论在金融和偏微分的概率解释中有重要应用价值的,由Levy过程驱动的倒向双重随机微分方程：通过证明上述方程的适应解在Malliavin微分意义下的可微性,给出了一种易于判断的比较定理的存在条件.第一节利用函数逼近的方法,证明了当终值ξ与系数函数满足一定条件时,倒向双重随机微分方程解的高阶矩的存在性(定理2.3.1,定理2.3.2).第二节和第三节,利用Malliavin微分的性质和Picard迭代法,分别证明了倒向双重随机微分方程对Wiener过程{Bt}的一阶、二阶Malliavin导数的存在性,指出解的一阶、二阶Malliavin导数也满足一个线性的倒向双重随机微分方程(定理2.4.1,定理2.5.1),同时也证明了倒向双重随机微分方程的解的一、二阶Malliavin导数的高阶矩的存在性(定理2.4.2,定理2.5.2).第四节,通过证明了带跳的Skorohod积分意义下的Ito公式(定理2.6.1)和Doleans-Dade公式(定理2.6.2),以及对生成Teugels鞅的Levy过程的详细讨论,给出了系数函数f与Z有关时,在“负跳一致小”条件下适应解的比较定理(定理2.6.4).
第三章讨论由Levy过程驱动的倒向随机微分方程在不同Bihari条件下,适应解的存在唯一性.第一节讨论由Levy过程驱动的倒向双重随机微分方程(Eq.(1)),当系数g满足Lipschitz条件,f满足推广的Bihari条件：|f(t,y1,u1,z1)-f(t,y2,u2,z2)|2≤c(t)k(|y1-y2|2)+K(|u1-u2|2+‖z1-z2‖2)时,利用推广It6公式、Picard迭代法和区间延拓过程,证明了上述方程Ft-适应解的存在唯一性(定理3.2.1).第二节利用推广的Bihari不等式和截断函数,给出了由Levy过程驱动的倒向随机微分方程在局部Bihari条件下解的存在唯一性(定理3.3.2).第三节同样借助推广的Bihari不等式和一些光滑函数,给出了由Levy过程驱动的倒向随机微分方程在单调Bihari条件下解的存在唯一性(定理3.4.1,定理3.4.2).
第四章引入变差逼近的思想,研究如下形式的随机微分方程的两点边值问题dXt=f(t,Xt)dt+σ(t,Xt)dWt,AX0+BXT=ξ*.给出适应解存在的充分必要条件(定理4.3.1).在f(t,Xt)=ft的简单情况下,通过放宽可行解范围引入控制项ft,把解Xt延拓为(Xt,ft),并利用构造随机序列的方法得到方程的一个解(定理44.1),同时也证明了所构造的解对边值的连续依赖性(定理4.5.1).最后把我们得到解的充要条件分别与Strum-Liouvelle问题(关于正交性条件的例4.6.1)以及Peng关于倒向随机微分方程的“鞅逼近”问题(关于布朗桥的例4.6.2)相比较,说明本文给出的变差适应解的广泛性.
第五章总结了本学位论文的主要内容,并给出了研究展望.
The purpose of this dissertation is to investigate the properties of the adapted solutions of several kinds of terminal value problems and boundary value problems of stochastic differential equations driven by three mutually independent processes:two continuous Wiener process Wt, Bt, and a Levy process Lt. The correspondingσ-algebra Ft is neither increasing nor decreas-ing, so it does not constitute a general filtration. There are three main contributions. Firstly, by proving the Ito formula and Doleans-Dade formula of the Skorohod integral with jump in the sense of general Skorohod integration, we show a comparison theorem of the backward doubly stochastic differential equation driven by Levy process when the coefficient function f is de-pendent on Z, and the sizes of all negative jumps are uniformly small. To our knowledge, this is the first time that such theorem is suggested. Secondly, we show the existence and uniqueness of the adapted solution of backward stochastic differential equations driven by Levy process under various Bihari conditions. Lastly, different from the martingale representation theorem of Pardoux and Peng, by introducing the idea of variational approximation the existence of the adapted solution of the two-point boundary value problem of the stochastic differential equa-tion is investigated, for which the sufficient and necessary condition is indicated. Meanwhile, an adapted solution of the underlying equation is obtained by constructing a solution sequences. The dissertation is organized as the following.
Chapter 1 gives some preliminaries for this dissertation, including some results and prop-erties of Levy process, Skorohod integral and Malliavin differential.
In Chapter 2, we study the backward doubly stochastic differential equations driven by Levy process: which have important applications in finance theory and the probability interpretation of partial differential equations. By discussing the Malliavin differentiability of the solution of the above equation, we draw a comparison theorem of which the assumed conditions can be checked very easily. By methods in approximation of functions, Section 2.1 proves the existence(Theorems 2.3.1 and 2.3.2) of the higher-order moments of the solutions of the backward doubly stochastic differential equations when the terminal valueξand the coefficient functions satisfy certain con-ditions. According to the properties of the Malliavin differential and Picard iteration, Sections 2.2 and 2.3 prove the existence of the first order and second order Malliavin derivatives of the backward doubly stochastic differential equation with respect to the Wiener process{Bt}, re-spectively. One-order and two-order Malliavin derivatives of the solutions are proved to be sat-isfied with a linear backward doubly stochastic differential equation(Theorems 2.4.1 and 2.5.1). Meanwhile, the existence(Theorems 2.4.2 and 2.5.2) of higher-order moments of the first or-der and second order Malliavin derivatives of the solutions of the backward doubly stochastic differential equations are proved respectively. In Section 2.4, the Ito formula(Theorem 2.6.1) and Doleans-Dade formula(Theorem 2.6.2) of the Skorohod integral with jump are proved, the Levy process from which the Teugels martingales are derived is discussed thoroughly. After that, we draw a new comparison theorem(Theorem 2.6.4) for the adapted solution in the sense of that the sizes of all negative jumps are uniformly small when the coefficient function f is dependent on Z.
The third chapter discusses the existence and uniqueness of the adapted solution of the backward stochastic differential equation driven by Levy process under various Bihari condi-tions. In Section 3.1, we consider the backward doubly stochastic differential equations driven by Levy process(Eq.(1)). When the coefficient g satisfies the Lipschitz condition, and f satis-fies the generalized Bihari condition: |f(t,y1,u1,z1)-f(t,y2,u2,z2)|2≤< c(t)k(|y1-y2|2)+K(|u1-u2|2+‖z1-z2‖2), we can prove the existence and uniqueness(Theorem 3.2.1) of the Ft-adapted solutions of Eq.(1) by the generalized Ito's formula, Picard iteration, and interval extension process. By the generalized Bihari inequality and truncation function, the existence and uniqueness(Theorem 3.3.2) of the solution of the backward tochastic differential equation driven by Levy process under local Bihari conditions are proved in Section 3.2. By the generalized Bihari inequality again and some smooth functions, Section 3.3 proves the same results(Theorem 3.4.1 and 3.4.2) but under monotony Bihari conditions.
By introducing the idea of variational approximation, Chapter 4 investigates the sufficient and necessary conditions(Theorem 4.3.1) for the existence of the adapted solutions of the two-point boundary value problems of the stochastic differential equation of the following form dXt=f(t, Xt)dt+σ(t, Xt)dWt. AX0+BXT=ξ*. In the simple case of f(t, Xt)=ft, a solution of this equation(Theorem 4.4.1) can be obtained by introducing a control term ft, extending the solution from Xt to (Xt,ft), and construct-ing a solution sequence. Meanwhile, we prove the continuous dependency(Theorem 4.5.1) of the constructed solution on the boundary value. Lastly, the sufficient and necessary con-ditions for the solution we obtained are compared with those of the Strum-Liouvelle prob-lems(Example 4.6.1), and the "martingale approximation" problem(Example 4.6.2) proposed by Peng for backward stochastic differential equations, which show the universality of our variational adapted solution.
Chapter 5 gives the conclusion and future work.
第一章给出本学位论文所需要的预备知识,包括Levy过程,Skorohod积分以及Malliavin微分的一些主要结论和性质.
第二章讨论在金融和偏微分的概率解释中有重要应用价值的,由Levy过程驱动的倒向双重随机微分方程：通过证明上述方程的适应解在Malliavin微分意义下的可微性,给出了一种易于判断的比较定理的存在条件.第一节利用函数逼近的方法,证明了当终值ξ与系数函数满足一定条件时,倒向双重随机微分方程解的高阶矩的存在性(定理2.3.1,定理2.3.2).第二节和第三节,利用Malliavin微分的性质和Picard迭代法,分别证明了倒向双重随机微分方程对Wiener过程{Bt}的一阶、二阶Malliavin导数的存在性,指出解的一阶、二阶Malliavin导数也满足一个线性的倒向双重随机微分方程(定理2.4.1,定理2.5.1),同时也证明了倒向双重随机微分方程的解的一、二阶Malliavin导数的高阶矩的存在性(定理2.4.2,定理2.5.2).第四节,通过证明了带跳的Skorohod积分意义下的Ito公式(定理2.6.1)和Doleans-Dade公式(定理2.6.2),以及对生成Teugels鞅的Levy过程的详细讨论,给出了系数函数f与Z有关时,在“负跳一致小”条件下适应解的比较定理(定理2.6.4).
第三章讨论由Levy过程驱动的倒向随机微分方程在不同Bihari条件下,适应解的存在唯一性.第一节讨论由Levy过程驱动的倒向双重随机微分方程(Eq.(1)),当系数g满足Lipschitz条件,f满足推广的Bihari条件：|f(t,y1,u1,z1)-f(t,y2,u2,z2)|2≤c(t)k(|y1-y2|2)+K(|u1-u2|2+‖z1-z2‖2)时,利用推广It6公式、Picard迭代法和区间延拓过程,证明了上述方程Ft-适应解的存在唯一性(定理3.2.1).第二节利用推广的Bihari不等式和截断函数,给出了由Levy过程驱动的倒向随机微分方程在局部Bihari条件下解的存在唯一性(定理3.3.2).第三节同样借助推广的Bihari不等式和一些光滑函数,给出了由Levy过程驱动的倒向随机微分方程在单调Bihari条件下解的存在唯一性(定理3.4.1,定理3.4.2).
第四章引入变差逼近的思想,研究如下形式的随机微分方程的两点边值问题dXt=f(t,Xt)dt+σ(t,Xt)dWt,AX0+BXT=ξ*.给出适应解存在的充分必要条件(定理4.3.1).在f(t,Xt)=ft的简单情况下,通过放宽可行解范围引入控制项ft,把解Xt延拓为(Xt,ft),并利用构造随机序列的方法得到方程的一个解(定理44.1),同时也证明了所构造的解对边值的连续依赖性(定理4.5.1).最后把我们得到解的充要条件分别与Strum-Liouvelle问题(关于正交性条件的例4.6.1)以及Peng关于倒向随机微分方程的“鞅逼近”问题(关于布朗桥的例4.6.2)相比较,说明本文给出的变差适应解的广泛性.
第五章总结了本学位论文的主要内容,并给出了研究展望.
The purpose of this dissertation is to investigate the properties of the adapted solutions of several kinds of terminal value problems and boundary value problems of stochastic differential equations driven by three mutually independent processes:two continuous Wiener process Wt, Bt, and a Levy process Lt. The correspondingσ-algebra Ft is neither increasing nor decreas-ing, so it does not constitute a general filtration. There are three main contributions. Firstly, by proving the Ito formula and Doleans-Dade formula of the Skorohod integral with jump in the sense of general Skorohod integration, we show a comparison theorem of the backward doubly stochastic differential equation driven by Levy process when the coefficient function f is de-pendent on Z, and the sizes of all negative jumps are uniformly small. To our knowledge, this is the first time that such theorem is suggested. Secondly, we show the existence and uniqueness of the adapted solution of backward stochastic differential equations driven by Levy process under various Bihari conditions. Lastly, different from the martingale representation theorem of Pardoux and Peng, by introducing the idea of variational approximation the existence of the adapted solution of the two-point boundary value problem of the stochastic differential equa-tion is investigated, for which the sufficient and necessary condition is indicated. Meanwhile, an adapted solution of the underlying equation is obtained by constructing a solution sequences. The dissertation is organized as the following.
Chapter 1 gives some preliminaries for this dissertation, including some results and prop-erties of Levy process, Skorohod integral and Malliavin differential.
In Chapter 2, we study the backward doubly stochastic differential equations driven by Levy process: which have important applications in finance theory and the probability interpretation of partial differential equations. By discussing the Malliavin differentiability of the solution of the above equation, we draw a comparison theorem of which the assumed conditions can be checked very easily. By methods in approximation of functions, Section 2.1 proves the existence(Theorems 2.3.1 and 2.3.2) of the higher-order moments of the solutions of the backward doubly stochastic differential equations when the terminal valueξand the coefficient functions satisfy certain con-ditions. According to the properties of the Malliavin differential and Picard iteration, Sections 2.2 and 2.3 prove the existence of the first order and second order Malliavin derivatives of the backward doubly stochastic differential equation with respect to the Wiener process{Bt}, re-spectively. One-order and two-order Malliavin derivatives of the solutions are proved to be sat-isfied with a linear backward doubly stochastic differential equation(Theorems 2.4.1 and 2.5.1). Meanwhile, the existence(Theorems 2.4.2 and 2.5.2) of higher-order moments of the first or-der and second order Malliavin derivatives of the solutions of the backward doubly stochastic differential equations are proved respectively. In Section 2.4, the Ito formula(Theorem 2.6.1) and Doleans-Dade formula(Theorem 2.6.2) of the Skorohod integral with jump are proved, the Levy process from which the Teugels martingales are derived is discussed thoroughly. After that, we draw a new comparison theorem(Theorem 2.6.4) for the adapted solution in the sense of that the sizes of all negative jumps are uniformly small when the coefficient function f is dependent on Z.
The third chapter discusses the existence and uniqueness of the adapted solution of the backward stochastic differential equation driven by Levy process under various Bihari condi-tions. In Section 3.1, we consider the backward doubly stochastic differential equations driven by Levy process(Eq.(1)). When the coefficient g satisfies the Lipschitz condition, and f satis-fies the generalized Bihari condition: |f(t,y1,u1,z1)-f(t,y2,u2,z2)|2≤< c(t)k(|y1-y2|2)+K(|u1-u2|2+‖z1-z2‖2), we can prove the existence and uniqueness(Theorem 3.2.1) of the Ft-adapted solutions of Eq.(1) by the generalized Ito's formula, Picard iteration, and interval extension process. By the generalized Bihari inequality and truncation function, the existence and uniqueness(Theorem 3.3.2) of the solution of the backward tochastic differential equation driven by Levy process under local Bihari conditions are proved in Section 3.2. By the generalized Bihari inequality again and some smooth functions, Section 3.3 proves the same results(Theorem 3.4.1 and 3.4.2) but under monotony Bihari conditions.
By introducing the idea of variational approximation, Chapter 4 investigates the sufficient and necessary conditions(Theorem 4.3.1) for the existence of the adapted solutions of the two-point boundary value problems of the stochastic differential equation of the following form dXt=f(t, Xt)dt+σ(t, Xt)dWt. AX0+BXT=ξ*. In the simple case of f(t, Xt)=ft, a solution of this equation(Theorem 4.4.1) can be obtained by introducing a control term ft, extending the solution from Xt to (Xt,ft), and construct-ing a solution sequence. Meanwhile, we prove the continuous dependency(Theorem 4.5.1) of the constructed solution on the boundary value. Lastly, the sufficient and necessary con-ditions for the solution we obtained are compared with those of the Strum-Liouvelle prob-lems(Example 4.6.1), and the "martingale approximation" problem(Example 4.6.2) proposed by Peng for backward stochastic differential equations, which show the universality of our variational adapted solution.
Chapter 5 gives the conclusion and future work.
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