The stochastic parabolic partial differential equation with non-Lipschitz coefficients on the unbounded domain

We consider the existence and uniqueness of the stochastic heat equation driven by the space-time white noise under the weaker conditions of the coefficients than the Lipschitz conditions. In order to show the existence of the solution, we investigate the sequence of the successive approximations under two kinds of conditions.     

The uniqueness of solutions of perturbed backward stochastic differential equations

We prove a result on the preservation of the pathwise uniqueness property for the adapted solution to backward stochastic differential equation under perturbations.     

由无穷个Brown单驱动的随机微分方程解的存在唯一性

考虑如下一维双参数随机微分方程: ,其中{Wj,j=1,2,…}为一列无穷个相互独立的实值Brown单．作者定义关于无穷个Brown单的随机积分,并给出方程在非Lipschitz系数的条件下解的存在唯一性的一个结果．     

On the strong and weak solutions of stochastic differential equations governing Bessel processes

We prove that the δ-dimensional Bessel process (δ > 1) is a strong solution of a stochastic differential equation of the special form. The purpose of this paper is to investigate whether there exist other (weak and strong) solutions of these equations. This leads us to the conclusion that Zvonkin's theorem cannot be extended to stochastic differential equations with an unbounded drift.     

A study of a class of stochastic differential equations with non-Lipschitzian coefficients

We study a class of stochastic differential equations with non-Lipschitz coefficients. A unique strong solution is obtained and the non confluence of the solutions of stochastic differential equations is proved. The dependence with respect to the initial values is investigated. To obtain a continuous version of solutions, the modulus of continuity of coefficients is assumed to be less than |x-y| log Finally a large deviation principle of Freidlin-Wentzell type is also established in the paper.     

一类(QL)型随机微分方程解的轨道唯一性判别及其应用

考虑了一类拟左连续(QL)型随机微分方程(S.D.E.)解的轨道唯一性,应用随机分析方法获得了唯一性成立的一般判别定理,并在方程系数满足局部(或非)Lipschitz条件下给出了一些应用实例.     

Well-posedness and long time behavior of singular Langevin stochastic differential equations

In this paper, we study damped Langevin stochastic differential equations with singular velocity fields. We prove the strong well-posedness of such equations. Moreover, by combining the technique of Lyapunov functions with Krylov’s estimate, we also establish exponential ergodicity for the unique strong solution.     

Uniqueness of Nash equilibrium for linear-convex stochastic differential games

The uniqueness of Nash equilibria is shown for a class of stochastic differential games where the dynamic constraints are linear in the control variables. The result is applied to an oligopoly.This paper benefitted from comments by two anonymous referees and by L. Blume and C. Simon.     

无穷可数个Brown运动驱动的随机微分方程解的分布唯一性及路径唯一性

研究如下形式的随机微分方程Xit =xi + ∑∞j=1∫t0σijs(Xs)dBjs +∫t0bis(Xs)ds,i=1,2,…,n,( * )其中{Bjt}∞j=1是相互独立的标准Brown运动的无穷可数序列.主要证明如下结论:1)解的分布唯一性蕴含了解的联合分布唯一性;2)解的分布唯一性与强解的存在性可以保证解的轨道唯一性.结论2)是Yamada定理的对偶命题.     

Banach值随机微分方程解的存在唯一性

本文我们用逐次逼近的方法研究Banach值随机微分方程解的存在性和轨道唯一性.     

Uniqueness and explosion time of solutions of stochastic differential equations driven by fractional Brownian motion

In this paper, we first study the existence and uniqueness of solutions to the stochastic differential equations driven by fractional Brownian motion with non-Lipschitz coefficients. Then we investigate the explosion time in stochastic differential equations driven by fractional Browmian motion with respect to Hurst parameter more than half with small diffusion.     

Uniqueness of positive solutions of a class of quasilinear ordinary differential equations

Abstract. Uniqueness results are obtained for positive solutions of a class of quasilinear ordi-nary differential equations. The methods rely on the energy analysis and a scale argument.     

Pathwise Uniqueness of the Solutions toVolterra Type Stochastic DifferentialEquations in the Plane

In this paper we prove the pathwise uniqueness of a kind of two-parameter Volterra type stochastic differential equations under the coefficients satisfy the non-Lipschitz conditions. We use a martingale formula in stead of Ito formula, which leads to simplicity the process of proof and extends the result to unbounded coefficients case.     

On pathwise uniqueness of stochastic evolution equations in Hilbert spaces

The pathwise uniqueness of stochastic evolution equations driven by Q-Wiener processes is mainly investigated in this article. We focus on the case that the modulus of the continuity of the coefficients is not controlled by a linear function. Additionally, we show that the corresponding diffusion process is Feller.     

On existence and uniqueness of solutions to uncertain backward stochastic differential equations简

This paper is concerned with a class of uncertain backward stochastic differential equations （UBSDEs） driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian coefficients. Such equations can be useful in mod- elling hybrid systems, where the phenomena are simultaneously subjected to two kinds of un- certainties： randomness and uncertainty. The solutions of UBSDEs are the uncertain stochastic processes. Thus, the existence and uniqueness of solutions to UBSDEs with Lipschitzian coeffi- cients are proved.     

Pathwise convergence rates for numerical solutions of Markovian switching stochastic differential equations

This work develops numerical approximation algorithms for solutions of stochastic differential equations with Markovian switching. The existing numerical algorithms all use a discrete-time Markov chain for the approximation of the continuous-time Markov chain. In contrast, we generate the continuous-time Markov chain directly, and then use its skeleton process in the approximation algorithm. Focusing on weak approximation, we take a re-embedding approach, and define the approximation and the solution to the switching stochastic differential equation on the same space. In our approximation, we use a sequence of independent and identically distributed (i.i.d.) random variables in lieu of the common practice of using Brownian increments. By virtue of the strong invariance principle, we ascertain rates of convergence in the pathwise sense for the weak approximation scheme.     

Uniqueness of positive solutions for Sturm-Liouville boundary value problems

Sufficient conditions for the uniqueness of positive solutions of singular Sturm-Liouville boundary value problems

where and , are established.

     

approximation for the solutions of stochastic differential equations. iii. jointly weak convergence

We give sufficient conditions for a family Z, e > 0 of continuous finite variation processes to converge weakly to a diffusion process Z. Then we consider the integral equation dXE(t) = (l)(Xe(t))dZE{t) and the stochastic equation dX{i) = (j)(X{t))dZ{t) and denote by X(t,x,w respectively X{t,x,(jo), the solution starting at x. We prove that PoX~l, e>0 converge weakly to Pol