由连续半鞅驱动的倒向随机微分方程解的比较定理

彭实戈[1]首先建立了一维倒向随机微分方程的比较定理,本文在Lipschitz条件下研究由连续半鞅驱动的倒向随机微分方程,我们将比较定理推广到此类倒向随机微分方程,并且证明方法比彭实戈[1]的更加直接和简单.     

非Lipschitz条件下的包含下微分算子的带跳倒向随机微分方程(英文)

本文在非Lipschitz系数下,考虑了一类多值的倒向随机微分方程.利用极大单调算子的Yosida估计和倒向随机微分方程在非Lipschitz条件下解的存在唯一性,获得了多值带跳的倒向随机微分方存在唯一解的结论.     

非Lipschitz系数McKean-Vlasov随机微分方程的大偏差原理

本文研究有限维框架下一类非Lipschitz系数McKean-Vlasov随机微分方程的Freidlin-Wentzell型大偏差原理,将此类条件下经典随机微分方程的相关结论推广到McKean-Vlasov随机微分方程.在此类McKean-Vlasov随机微分方程解的存在唯一性基础上,采用弱收敛方法得到其大偏差原理.     

一类非Lipschitz条件的Backward SDE适应解的存在唯一性

本文中，我们在非Lipschitz条件下证明了倒向随机微分方程的局部与整体适应解的存在唯一性，推广了PaLrdoux-Peng定理．     

平均场倒向重随机微分方程及其应用

研究了平均场倒向随重机微分方程, 得到了平均场倒向重随机微分方程解的存在唯一性.基于平均场倒向重随机微分方程的解, 给出了一类非局部随机偏微分方程解的概率解释.讨论了平均场倒向重随机系统的最优控制问题, 建立了庞特利亚金型的最大值原理.最后讨论了一个平均场倒向重随机线性二次最优控制问题, 展示了上述最大值原理的应用.     

一般正倒向重随机微分方程的解

研究了一类正倒向重随机微分方程,其涵盖了以前的包括随机Hamilton系统的很多情况.通过连续性方法,在较弱的单调条件下得到了其解的存在唯一性结果.然后研究了正倒向重随机微分方程的解依赖于参数的连续性和可微性.     

非连续生成元倒向重随机微分方程最小解的存在性及比较定理

本文研究生成元f关于(y,z)线性增长,关于y左连续和左Lipschitz且关于z连续的一维倒向重随机微分方程,证明其最小解的存在性,得到其最小解的一个比较定理,推广了几个已有结果.     

多维带跳倒向双重随机微分方程解的性质

本文研究一类多维带跳倒向双重随机微分方程, 给出了It\^{o}公式在带跳倒向双重随机积分情形下的推广形式, 同时运用推广形式的It\^{o}公式, 在Lipschitz条件下证明了方程解的存在性和唯一性.     

分数布朗运动驱动下z一致连续的BSDE解的存在性与唯一性

本文研究一类由分数布朗运动驱动的一维倒向随机微分方程解的存在性与唯一性问题,在假设其生成元满足关于y Lipschitz连续,但关于z一致连续的条件下,通过应用分数布朗运动的Tanaka公式以及拟条件期望在一定条件下满足的单调性质,得到倒向随机微分方程的解的一个不等式估计,应用Gronwall不等式得到了一个关于这类方程的解的存在性与唯一性结果,推广了一些经典结果以及生成元满足一致Lipschitz条件下的由分数布朗运动驱动的倒向随机微分方程解的结果.     

带随机初值的随机偏微分方程整体解的存在性

研究了一类带随机初值并且由分数次Brownian运动驱动的随机偏微分方程.借助于Kolmogorov准则,建立了整体Lipschitz条件下此类随机偏微分方程的一个解.同时证明了局部Lipschitz条件下整体解的存在性.     

A comparison theorem and uniqueness theorem of backward doubly stochastic differential equations

In this paper, we deal with one dimensional backward doubly stochastic differential equations (BDSDEs). We obtain a comparison theorem and a uniqueness theorem for BDSDEs with continuous coefficients.     

Backward doubly stochastic differential equations with weak assumptions on the coefficients

In this paper, we deal with one dimensional backward doubly stochastic differential equations (BDSDEs). We obtain existence theorems and comparison theorems for solutions of BDSDEs with weak assumptions on the coefficients.     

A generalized existence theorem of backward doubly stochastic differential equations

In this paper, we deal with a class of one-dimensional backward doubly stochastic differential equations （BDSDEs）. We obtain a generalized comparison theorem and a generalized existence theorem of BDSDEs.     

Stochastic viscosity solutions for SPDEs with continuous coefficients

In this note, nonlinear stochastic partial differential equations (SPDEs) with continuous coefficients are studied. Via the solutions of backward doubly stochastic differential equations (BDSDEs) with continuous coefficients, we provide an existence result of stochastic viscosity sub- and super-solutions to this class of SPDEs. Under some stronger conditions, we prove the existence of stochastic viscosity solutions.     

Stationary solutions of SPDEs and infinite horizon BDSDEs

In this paper we study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between valued solutions of backward doubly stochastic differential equations (BDSDEs) on infinite horizon and the stationary solutions of the SPDEs. Moreover, we prove the existence and uniqueness of the solutions of BDSDEs on both finite and infinite horizons, so obtain the solutions of initial value problems and the stationary solutions (independent of any initial value) of SPDEs. The connection of the weak solutions of SPDEs and BDSDEs has independent interests in the areas of both SPDEs and BSDEs.     

A class of backward doubly stochastic differential equations with discontinuous coefficients

In this work the existence of solutions of one-dimensional backward doubly stochastic differential equations （BDSDEs） with coefficients left-Lipschitz in y （may be discontinuous） and Lipschitz in z is studied. Also, the associated comparison theorem is obtained.     

Infinite horizon backward doubly stochastic differential equations with non-degenerate terminal functions and their stationary property

In this paper we consider infinite horizon backward doubly stochastic differential equations (BDSDEs for short) coupled with forward stochastic differential equations, whose terminal functions are non-degenerate. For such kind of BDSDEs, we study the existence and uniqueness of their solutions taking values in weighted L ^p (dx)?L ²(dx) space (p ≥ 2), and obtain the stationary property for the solutions.     

Lp Solutions for Multidimensional BDSDEs with Locally Weak Monotonicity Coefficients

In this paper, the authors establish the existence and uniqueness theorem of L^p (1 < p ≤ 2) solutions for multidimensional backward doubly stochastic differential equations (BDSDEs for short) under the p-order globally (locally) weak monotonicity conditions. Comparison theorem of L^p solutions for one-dimensional BDSDEs is also proved.These conclusions unify and generalize some known results.     

正倒向随机微分方程理论及应用

正倒向随机微分方程源于随机控制和金融等问题的研究,反之,方程理论的研究成果在控制、金融等领域也有着重要的应用。基于正向和倒向随机微分方程的理论成果,正倒向随机微分方程的研究在短时间内取得了长足进步。本文将从方程可解性这一角度出发,对正倒向随机微分方程目前取得的成果进行系统的总结与探讨。     

Backward stochastic Volterra integral equations—Representation of adapted solutions

For backward stochastic Volterra integral equations (BSVIEs, for short), under some mild conditions, the so-called adapted solutions or adapted M-solutions uniquely exist. However, satisfactory regularity of the solutions is difficult to obtain in general. Inspired by the decoupling idea of forward–backward stochastic differential equations, in this paper, for a class of BSVIEs, a representation of adapted M-solutions is established by means of the so-called representation partial differential equations and (forward) stochastic differential equations. Well-posedness of the representation partial differential equations are also proved in certain sense.