Forward-backward Stochastic Differential Equations and Backward Linear Quadratic Stochastic Optimal Control Problem

In this paper, we use the solutions of forward-backward stochastic differential equations to get the optimal control for backward stochastic linear quadratic optimal control problem. And we also give the linear feedback regulator for the optimal control problem by using the solutions of a group of Riccati equations.     

倒向双重随机微分方程

本文研究了如下倒向随机微分方程Yt=ξ ∫t^Tf(x,Yt,Zt)ds ∫t^TB(ds,g(s,Yt,Zt))-∫t^TZtdW,, 在类似于Yamada条件下，得到了它解的存在唯一性定理，推广了Anis Matoussi和Michael Scheutzow相关结果．拓展倒向随机微分方程在随机控制问题和数理金融等方面的应用。     

Backward Doubly Stochastic Differential Equations with Jumps and Stochastic Partial Differential-Integral Equations

Backward doubly stochastic differential equations driven by Brownian motions and Poisson process(BDSDEP) with non-Lipschitz coeffcients on random time interval are studied.The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations(SPDIEs) is treated with BDSDEP.Under non-Lipschitz conditions,the existence and uniqueness results for measurable solutions to BDSDEP are established via the smoothing technique.Then,the continuous dependence for solutions to BDSDEP is derived.Finally,the probabilistic interpretation for the solutions to a class of quasilinear SPDIEs is given.     

系数为左Lipschitz的倒向随机微分方程解的存在性

考虑一类一维倒向随机微分方程(BSDE),其系数关于y满足左Lipschitz条件(可能是不连续的),关于z满足Lipschitz条件.在这样的条件下,证明了BSDE的解是存在的,并且得到了相应的比较定理.     

系数为左Lipschitz的倒向随机微分方程解的存在性

考虑一类一维倒向随机微分方程(BSDE),其系数关于y满足左Lipschitz条件(可能是不连续的),关于z满足Lipschitz条件.在这样的条件下,证明了BSDE的解是存在的,并且得到了相应的比较定理.     

Jensen's Inequality for Backward Stochastic Differential Equations

Under the Lipschitz assumption and square integrable assumption on g, the author proves that Jensen's inequality holds for backward stochastic differential equations with generator g if and only if g is independent of y, g(t, 0) = 0 and g is super homogeneous with respect to z. This result generalizes the known results on Jensen's inequality for g-expectation in [4, 7-9].     

Backward Stochastic Differential Equations in the Plane

This paper is devoted to study backward stochastic differential equations in the plane driven by a Brownian sheet, where the value of the solution at the corner (s ₀,t ₀) is fixed. The existence and uniqueness of a solution is obtained by means of Picard's approximation scheme and a suitable two-parameter Gronwall's type lemma.     

Two-Step Scheme for Backward Stochastic Differential Equations

In this paper, a stochastic linear two-step scheme has been presented to approximate backward stochastic differential equations (BSDEs). A necessary and sufficient condition is given to judge the $\mathbb{L}_2$-stability of our numerical schemes. This stochastic linear two-step method possesses a family of $3$-order convergence schemes in the sense of strong stability. The coefficients in the numerical methods are inferred based on the constraints of strong stability and $n$-order accuracy ($n\in\mathbb{N}^+$). Numerical experiments illustrate that the scheme is an efficient probabilistic numerical method.     

The Link between Stochastic Differential Equations with Non-Markovian Coefficients and Backward Stochastic Partial Differential Equations

In this paper, we conjecture and prove the link between stochastic differential equations with non-Markovian coefficients and nonlinear parabolic backward stochastic partial differential equations, which is an extension of such kind of link in Markovian framework to non-Markovian framework.Different from Markovian framework, where the corresponding partial differential equation is deterministic, the backward stochastic partial differential equation here has a pair of adapted solutions, and thus the link has a much different form. Moreover, two examples are given to demonstrate the applications of the derived link.     

倒向随机微分方程及其应用

本文将介绍一类新的议程：倒向随机微分方程，为了便于理解，我们将首先通过与常微分方程和经典的随机微分方程的对比，并通过数理经济和数学金融学中的一个典型的例子来引入倒向随机微分方程。     

Exponential Stability for Time-changed Stochastic Differential Equations

Acta Mathematicae Applicatae Sinica, English Series - So far there have been few results presented on the exponential stability for time-changed stochastic differential equations. The main aim of...     

Markov调制奇异随机微分方程的指数稳定性

建立了Markov调制奇异随机微分方程的p阶指数稳定性和几乎必然指数稳定性的充要条件．     

倒向随机微分方程弱解（英）

引入倒向随机微分方程弱解的概念，应用Girsanov变换，建立了两类倒向随机微分方程（0．1）和（0．2）弱解存在的等价性，由此得到倒向 随机微分方程弱解存在的几个充分条件。     

Numerical Method for Reflected Backward Stochastic Differential Equations

In this article we propose a numerical method for reflected backward stochastic differential equations (RBSDE). This method is based on the simple random walk, and the convergence is related to the Skorohod topology.     

高维倒向随机微分方程比较定理

倒向随机微分方程由Pardoux和彭实戈首先提出，彭实戈给出了一维BSDE的比较定理，周海滨将其推广到了高维情形．毛学荣将倒向随机微分方程解的存在唯一性定理推广到非Lipschitz系数情况，曹志刚和严加安给了相应的一维比较定理．本文将曹志刚和严加安的比较定理推广到高维情形．     

非Lipschitz条件下的倒向重随机微分方程

该文研究了非Lipschitz条件下的倒向重随机微分方程, 给出了此类方程解的存在唯一性 定理, 推广Pardoux和Peng 1994年的结论; 同时也得到了此类方程在非Lipschitz条件下的比较定理, 推广了Shi,Gu和Liu 2005年的结果. 从而推广倒向重随机微分方程在随机控制和随机偏微分方程在 粘性解方面的应用.     

倒向随机微分方程的理论、发展及其应用

本文全面综述了倒向随机微分方程理论的出现、发展、应用及研究现状，介绍了作者博士论文的主要工作。     

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.