【作者】 吴浩；

【导师】 张希承；

【作者基本信息】 武汉大学 ， 概率论与数理统计， 2013， 博士

【摘要】 这篇论文以若干不同类型的倒向随机微分方程以及其应用为主要研究内容,包含了第二章,我们减弱了Peng和Yang [76]这篇文章中生成子的条件,得到了延迟倒向随机稳分方程解的存在唯一性、比较定理以及这类方程的LP解。在求Lp解的过程中,和[21]中求Lp解的方法比较,我们运用更直接的Picard迭代方法。第三章中,我们研究了非Lipschitz条件下带跳的延迟倒向随机微分方程,获得了这种方程和带跳的延迟随机微分方程之间一种新的对偶关系,存在唯一性定理,比较定理及其这类方程的一种新的变形方程。第四章中,我们研究了重反射广义延迟倒向随机微分方程,得到了这种方程解的存在唯一性定理和带有可加扰动解的稳定性,同时通过这种方程解的存在唯一性,我们得到带有泛函障碍的重反射广义倒向随机微分方程解的存在唯一性以及把这类带有泛函障碍的重反射广义倒向随机微分方程运用到一个最优停时问题。最后一章,我们把均场倒向随机微分方程,重倒向随机微分方程和反射倒向随机微分方程结合起来,研究了均场反射重倒向随机微分方程,得到解的存在唯一性和比较定理,并且通过这种方程的比较定理得到了这类方程的最小解和最大解。在求最小解和最大解的过程中,改变了通常要求倒向随机微分方程的生成子满足连续性条件和线性增长的条件,在生成子非连续和生成子满足类似线性增长的条件下,我们得到了这种方程的最小解和最大解。特别地,我们修改了[28]中引理3.1的第(V)条证明中的一个小瑕疵,我们将生成子的线性增长性换成一种类似线性增长性后,这个小瑕疵就可以得到解决。在这种方程中,生成子如果满足通常线性增长条件,将得不到方程的最大解和最小解的结论,当我们把f的线性增长性条件改变后,运用一种不同的证明方法就可以避免这种瑕疵。具体的[28]中引理3.1的小瑕疵及解决方法将在本文主要内容中详细说明。更多还原

【Abstract】 This thesis is devoted to the studies of several classes of backward stochastic differ-ential equations and their applications.In the first Chapter, we give an overview about BSDEs for the reader’s convenience. In particular, recall some well-known results.In Chapter2, we weaken the Lipschitz conditions on generators for anticipated backward stochastic differential equations in Peng and Yang [76]. By using Bihari’s inequality, we obtain the existence and uniqueness result for this type of equations and comparison theorems under non-Lipschitz assumptions. Furthermore, we use the more direct Picard’s iteration argument to get the Lp solutions compared with the method used in [21].In Chapter3, basing on [76], we study anticipated backward stochastic differential equations with jumps under non-Lipschitz conditions. We get a dual relation between stochastic delay differential equations and this type of equations. Moreover, we also obtain the existence and uniqueness result, Lp solutions for this type of equations, using a different method compared with the method used in Chapter2and with the different assumptions, we get the comparison theorem and its lemma. At last, we study a new type of anticipated backward stochastic differential equations and get the existence and uniqueness result.In Chapter4, combining the generalized anticipated backward stochastic differential equations with doubly reflected backward stochastic differential equations, we study the doubly reflected generalized anticipated backward stochastic differential equations and obtain the existence and uniqueness result as well as stability of solutions. Then, we give several applications for this type of equation. Using the existence and uniqueness result of this type of equations, we study the existence and uniqueness results of doubly reflected generalized anticipated backward stochastic differential equations with two functional barriers, which is applied to stochastic games.In Chapter5, by studying mean field backward stochastic differential equations, backward doubly stochastic differential equations and reflected backward stochastic dif-ferential equations, we study mean field reflected backward doubly stochastic differential equations, and obtain the existence and uniqueness results as well as the comparison theorems. As an application of comparison theorem of this type equations, we get its responding minimal solution and maximal solution. In the general, we often require that the generators should be satisfied the continuous condition and linear growth condition. Compared with the usual conditions, we can obtain similar results, which we don’t re-quire the generators satisfy the continuous conditions. In particular, We modify a minor fault in the proof of (V)in Lemma3.1of [28]. By changing the linear condition of f into a similar the linear condition, we can solve this minor fault. If f satisfies the usual linear growth condition, we can not get the the minimal solution and maximal solution results for this type of equations. When changing the linear condition of f into a similar the linear condition, we can use a different method to finish the proof. The concrete method will be described in the main results of this chapter.更多还原