A Comparison Theorem for Solution of the Fully Coupled Backward Stochastic Differential Equations

The comparison theorems of solutions for BSDEs in fully coupled forward-backward stochastic differential equations (FBSDEs) are studied in this paper, here in the fully coupled FBSDEs the forward SDEs are the same structure.

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.

SOME PROPERTIES OF SOLUTIONS OF ONE-DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH GENERAL COEFFICIENTS

In this paper, the one-dimensional time-homogenuous lto’s stochastic differential equations, which have degenerate and discontinuous diffusion coefficients, are considered. The non-confluent property of solutions is showed under some local integrability condition on the diffusion and drift coefficients. The strong comparison theorem for solutions is also established.

Relationship Among Solutions of a Generalized Riccati Equation

In this paper, some solutions of a generalized Riccati equation are investigated, which are given in the recent articles [Chaos, Solitons ＆ Fractals 24 （2005） 257; Phys. Lett. A 336 （2005） 463], and the relationship among the solutions is revealed.

A General Converse Comparison Theorem for Backward Stochastic Differential Equation with Non-lipschitz Coefficient

In this article, we first introduce g-expectation via the solution of backward stochastic differential equation（BSDE in short） with non-Lipschitz coefficient, and give the properties of g-expectation, then we establish a general converse comparison theorem for backward stochastic differential equation with non-Lipschitz coefficient.

A Limit Theorem for Solutions of Backward Stochastic Differential Equations

A limit theorem for solutions of backward stochastic differential equations was established. It extends aresult of Briand et al.

New Generalized (G'/G)-Expansion Method Applications to Coupled Konno-Oono Equation

The new generalized (G'/G)-expansion method is one of the powerful and competent methods that appear in recent time for establishing exact solutions to nonlinear evolution equations (NLEEs). We apply the new generalized (G'/G)-expansion method to solve exact solutions of the new coupled Konno-Oono equation and construct exact solutions expressed in terms of hyperbolic functions, trigonometric functions, and rational functions with arbitrary parameters. The significance of obtained solutions gives credence to the explanation and understanding of related physical phenomena. As a newly developed mathematical tool, this method efficiency for finding exact solutions has been demonstrated through showing its straightforward nature and establishing its ability to handle nonlinearities prototyped by the NLEEs whether in applied mathematics, physics, or engineering contexts.

Interval Criteria for Oscillation of Forced Second-order Half-linear Differential Equations

By employing a generalized Riccati technique and an integral averaging tech-nique, new interval oscillation criteria are established for the forced second-order half-lineardifferential equation [r(t)|x′ (t)|α-1x′ (t)]′ + q(t)|x(t)|α-1x(t) = e(t).

One Kind of Fully Coupled Linear Quadratic Stochastic Control Problem with Random Jumps

有随机的一个种有点线性的二次的随机的控制问题跳被学习。最佳的控制的明确的形式被获得。最佳的控制能被证明唯一。一个种概括 Riccati 方程系统被介绍，它的解决之可能性被讨论。为有随机的最佳的控制问题的线性反馈管理者跳被概括 Riccati 方程系统的解决方案给。

Conservation Laws of the Differential-Difference KP Equation

An infinite number of semi-discrete and continuous conservation laws for the differential-difference KP equation were obtained by using a solvable generalized Riccati equation.

New Interval Oscillation Criteria for Forced Second-order Half-linear Differential Equations

This paper discusses a class of forced second-order half-linear differential equations. By using the generalized Riccati technique and the averaging technique, some new interval oscillation criteria are obtained.

New Oscillation Results for Forced Second Order Differential Equations with Mixed Nonlinearities

Some new oscillation criteria are given for forced second order differential equations with mixed nonlinearities by using the generalized variational principle and Riccati technique. Our results generalize and extend some known oscillation results in the literature.

生成扩散模型研究综述

扩散模型在生成模型领域具有高质量的样本生成能力,一经推出就不断地刷新图像生成评价指标FID分数的记录,成为了该领域的研究热点,而此类相关综述在国内还鲜有介绍。因此,文中对相关扩散生成模型的研究进行汇总与分析。首先,对去噪扩散概率模型、基于分数的扩散生成模型和随机微分方程的扩散生成模型这3类通用模型的特点和原理进行了论述,就每一类基本扩散模型中以优化模型内部算法、高效采样为改进目标的相关衍生模型进行分析。其次,对当下扩散模型在计算机视觉、自然语言处理、时间序列、多模态和跨学科领域等方面的应用进行总结。最后,基于上述论述,分别就目前扩散生成模型存在的采样步骤多、采样时间长等局限性提出了相关建议,并结合前述研究对未来扩散生成模型的发展方向进行了研判。

一类Riccati微分方程通解的充要条件

文章讨论一类Riccati微分方程通解存在的充分必要条件,为求Riccati微分方程的解,提供一种有效方法,并给出它的应用.

函数变换法在Riccati方程求解中的应用

对Riccati方程进行函数变换,使之成为缺少一次项的特殊的Riccati方程,之后用已经成熟的理论进行进一步求解。

关于一类Riccati方程可积性条件的注记

基于Riccati方程的双参数特解,利用变量变换的思想,把一类与Riccati方程特解有关的可积性结果统一起来,并加以推广,得到了Riccati方程的一个更广泛的可积条件及其在该条件下的通积分。

利用微分算子法研究二阶齐次线性微分方程与Riccati方程通解之联系

文章应用微分算子法处理二阶变系数线性微分方程,揭示了二阶齐次变系数线性微分方程与Riccati方程的通解理论之间的联系,发现这两类方程之间的通解可以互相转化,同时给出转化的途径.最后,作为理论的应用分析了一些具体例子.

Riccati方程的广义反射函数与周期解

为了解决Riccati方程的周期解与稳定性,提出了通过Riccati方程的广义反射函数来寻找其Poincaré映射的方法。结果表明:从微分系统的广义反射函数角度出发,得到了Riccati方程具有线性广义反射函数的条件,从而推出了Riccati方程的线性广义反射函数,得到了Riccati方程的Poincaré映射,以及Riccati方程有周期解的条件与稳定性。由于通过广义反射函数寻找Poincaré映射比较简单可行,所以,Riccati方程的周期解与稳定性的研究成果对其它微分系统的研究也具有指导作用。

二阶变系数线性微分方程的Riccati方程解法

在(b′(x)b+2a(x)b(x))/b^2(x)≡c(常数)条件下,给出了微分方程y″+a(x)y′+b(x)y=f(x)(1)相对应的Riccati方程z′=z^2-a(x)z+b(x)(2)存在通解公式,进而得出了微分方程(1)或其齐次方程的通解公式.应用这些只与方程系数a(x)与b(x)相关的求解公式,求其通解过程十分简捷.

Uniqueness Problem for the Backward Differential Equation of a Continuous-State Branching Process

The distributional properties of a multi-dimensional continuous-state branching process are determined by its cumulant semigroup,which is defined by the backward differential equation.We provide a proof of the assertion of Rhyzhov and Skorokhod(Theory Probab.Appl.,1970)on the uniqueness of the solutions to the equation,which is based on a characterization of the process as the pathwise unique solution to a system of stochastic equations.