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.     

非全局Lipschitz条件下随机延迟微分方程Euler方法的收敛性

大多数随机延迟微分方程数值解的结果是在全局Lipschitz条件下获得的.许多延迟方程不满足全局Lipschitz条件,研究非全局Lipschitz条件下的数值解的性质,具有重要的意义.本文证明了漂移系数满足单边Lipschitz条件和多项式增长条件,扩散系数满足全局Lipschitz条件的一类随机延迟微分方程的Eul...     

无限时滞随机泛函微分方程的Razumikhin型定理

在无限时滞的随机泛函微分方程整体解存在的前提下,建立了一般衰减稳定性的Razumikhin型定理.在此基础上,基于局部Lipschitz条件和多项式增长条件,得到了无限时滞随机泛函微分方程整体解的存在唯一性,以及具有一般衰减速率的p阶矩和几乎必然渐近稳定性定理.     

非Lipschitz条件下半鞅随机微分方程解的唯一性(英文)

本文研究了非Lipschitz条件下半鞅随机微分方程.利用It(o)分析和Gronwall不等式,探讨了随机微分方程无爆炸解,并证明了随机微分方程解的唯一性.     

非Lipschitz系数下随机泛函微分方程适度解的存在唯一性及其渐近性态(英文)

本文主要运用Picard迭代和算子分数次幂方法,讨论了随机时滞偏微分方程适度解的存在性与唯一性,并对解的渐近性态进行了研究.这里方程的系数不满足Lipschitz条件,时滞r>0为有限的.最后给出了一个非Lipschitz条件的例子.     

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.     

中立型随机泛函微分方程的Khasminskii型定理

本文对中立型随机泛函微分方程建立了Khasminskii型定理,这个定理显示在局部Lipschitz条件但是不要求线性增长的条件下,中立型随机泛函微分方程存在一个全局解.本文的这个解存在性条件可以包含更广的一类非线性中立型随机泛函微分方程.最后,本文给出一个例子来阐述我们的思想.     

非Lipschitz条件下高维McKean-Vlasov随机微分方程解的存在唯一性

研究了一类漂移系数不连续的高维McKean-Vlasov随机微分方程及相应的粒子系统解的存在唯一性.在漂移系数关于空间变量逐段Lipschitz连续的条件下，首先利用Zvonkin变换将方程转换为漂移系数为Lipschitz连续的McKean-Vlasov随机微分方程，变换后的方程存在唯一解.然后由变换函数的性质可得逆函数的存在性和Lipschitz连续性.最后由It8公式及逆函数的性质可得原来的McKean-Vlasov随机微分方程及相应的粒子系统解的存在唯一性.     

一类以鞅为驱动的随机泛函微分方程强解的存在性与唯一性

在积分型Lipschitz条件下,证明了一类以连续鞅为驱动的随机泛函微分方程解的存在性与唯一性.     

Pathwise Numerical Approximations of SPDEs with Additive Noise under Non-global Lipschitz Coefficients

We consider the pathwise numerical approximation of nonlinear parabolic stochastic partial differential equations (SPDEs) driven by additive white noise under local assumptions on the coefficients only. We avoid the standard global Lipschitz assumption in the literature on the coefficients by first showing convergence under global Lipschitz coefficients but with a strong error criteria and then by applying a localization technique for one sample path on a bounded set.     

Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations Driven by Space-Time White Noise I

We approximate quasi-linear parabolic SPDEs substituting the derivatives in the space variable with finite differences. When the nonlinear terms in the equation are Lipschitz continuous we estimate the rate of L^p convergence of the approximations and we also prove their almost sure uniform convergence to the solution. When the nonlinear terms are not Lipschitz continuous we obtain this convergence in probability, if the pathwise uniqueness for the equation holds.     

On a class of Cahn–Hilliard type stochastic interacting systems with stepping-stone noises

We are concerned with a class of Cahn–Hilliard type stochastic interacting systems with stepping-stone noises. We first establish approximating SPDEs for them since the diffusion coefficients are not Lipschitz. And then we obtain the existence of weak mild solution to this system by solving a martingale problem.     

On non-degenerate quasi-linear stochastic partial differential equations

We prove a limit theorem for non-degenerate quasi-linear parabolic SPDEs driven by space-time white noise in one space-dimension, when the diffusion coefficient is Lipschitz continuous and the nonlinear drift term is only measurable. Hence we obtain an existence and uniqueness and a comparison theorem, which generalize those in [2], [4], [5] to the case of non-degenerate SPDEs with measurable drift and Lipschitz continuous diffusion coefficients.Research supported by the Hungarian National Foundation of Scientific Research No. 2290.     

STRONG CONVERGENCE OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR A CLASS OF SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH MULTIPLICATIVE NOISE

This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations(SPDEs)with multiplicative noise.The nonlinearity in the diffusion term of the SPDEs is assumed to be globally Lipschitz and the nonlinearity in the drift term is only assumed to satisfy a one-sided Lipschitz condition.These assumptions are the same ones as the cases where numerical methods for general nonlinear stochastic ordinary differential equations(SODEs)under"minimum assumptions"were studied.As a result,the semilinear SPDEs considered in this paper are a direct generalization of these nonlinear SODEs.There are several difficulties which need to be overcome for this generalization.First,obviously the spatial discretization,which does not appear in the SODE case,adds an extra layer of difficulty.It turns out a spatial discretization must be designed to guarantee certain properties for the numerical scheme and its stiffness matrix.In this paper we use a finite element interpolation technique to discretize the nonlinear drift term.Second,in order to prove the strong convergence of the proposed fully discrete finite element method,stability estimates for higher order moments of the H1-seminorm of the numerical solution must be established,which are difficult and delicate.A judicious combination of the properties of the drift and diffusion terms and some nontrivial techniques is used in this paper to achieve the goal.Finally,stability estimates for the second and higher order moments of the L2-norm of the numerical solution are also difficult to obtain due to the fact that the mass matrix may not be diagonally dominant.This is done by utilizing the interpolation theory and the higher moment estimates for the H1-seminorm of the numerical solution.After overcoming these difficulties,it is proved that the proposed fully discrete finite element method is convergent in strong norms with nearly optimal rates of convergence.Numerical experiment results are also presented to validate the theoretical results and to demonstrate the efficiency of the proposed numerical method.     

Existence-Uniqueness Problems For Infinite Dimensional Stochastic Differential Equations With Delays

The main aim of this paper is to develop the basic theory of a class of infinite dimensional stochastic differential equations with delays (IDSDEs) under local Lipschitz conditions. Firstly, we establish a global existence-uniqueness theorem for the IDSDEs under the global Lipschitz condition in \(C\) without the linear growth condition. Secondly, the non-continuable solution for IDSDEs is given under the local Lipschitz condition in \(C\). Then, the classical Itô's formula is improved and a global existence theorem for IDSDEs is obtained. Our new theorems give better results while conditions imposed are much weaker than some existing results. For example, we need only the local Lipschitz condition in \(C\) but neither the linear growth condition nor the continuous condition on the time \(t\). Finally, two examples are provided to show the effectiveness of the theoretical results.     

Finding the Minimal Root of an Equation with the Multiextremal and Nondifferentiable Left-Hand Part

A problem very often arising in applications is presented: finding the minimal root of an equation with the objective function being multiextremal and nondifferentiable. Applications from the field of electronic measurements are given. Three methods based on global optimization ideas are introduced for solving this problem. The first one uses an a priori estimate of the global Lipschitz constant. The second method adaptively estimates the global Lipschitz constant. The third algorithm adaptively estimates local Lipschitz constants during the search. All the methods either find the minimal root or determine the global minimizers (in the case when the equation under consideration has no roots). Sufficient convergence conditions of the new methods to the desired solution are established. Numerical results including wide experiments with test functions, stability study, and a real-life applied problem are also presented.     

Further results on existence-uniqueness for stochastic functional differential equations

The aim of this paper is to develop some basic theories of stochastic functional differential equations (SFDEs) under the local Lipschitz condition in continuous functions space ?. Firstly, we establish a global existence-uniqueness lemma for the SFDEs under the global Lipschitz condition in ? without the linear growth condition. Then, under the local Lipschitz condition in ?, we show that the non-continuable solution of SFDEs still exists if the drift coefficient and diffusion coefficient are square-integrable with respect to t when the state variable equals zero. And the solution of the considered equation must either explode at the end of the maximum existing interval or exist globally. Furthermore, some more general sufficient conditions for the global existence-uniqueness are obtained. Our conditions obtained in this paper are much weaker than some existing results. For example, we need neither the linear growth condition nor the continuous condition on the time t. Two examples are provided to show the effectiveness of the theoretical results.     

Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations driven by Space-Time White Noise II

We approximate quasi-linear parabolic SPDEs substituting the derivatives with finite differences. We investigate the resulting implicit and explicit schemes. For the implicit scheme we estimate the rate of L^p convergence of the approximations and we also prove their almost sure convergence when the nonlinear terms are Lipschitz continuous. When the nonlinear terms are not Lipschitz continuous we obtain convergence in probability provided pathwise uniqueness for the equation holds. For the explicit scheme we get these results under an additional condition on the mesh sizes in time and space.