巴拿哈空间中随机年龄结构种群方程的数值分析

本文研究年龄结构随机种群方程的离散误差,在空间离散中用到Galerkin公式,时间离散中用到显式欧拉公式.     

马尔可夫调制及带Poisson跳随机时滞微分方程依分布稳定

本文讨论马尔可夫调制及带Poisson跳随机时滞微分方程,其主要目的是研究方程解的依分布稳定.     

带Poisson跳非线性随机种群动力学模型的最优收获控制

研究了一类带跳的非线性随机群体动力学模型的最优收获控制.给出了在外界环境对系统产生影响的条件下带有Poisson跳的随机种群动力学系统;通过随机极大值原理,Hamilton函数及Ito公式,讨论了最优收获控制所满足的充分必要条件,所得到的结论是确定性种群系统的扩展.     

带Poisson跳随机微分方程终值与边值问题的适应解

Poisson跳的拟线性倒向随机微分方程x(t) ∫tf(s,x(s),,x(s)) y(s)]dMs =ξ,t∈[0,1],这里M = (W,Q)T,其中W为Wiener过程,Q为补偿Poisson过程.利用区间延拓和 Bihari 不等式证明了在某种弱于Lipschitz条件下方程存在唯一适应解,并给出了解的估计,从而将文章[1]的结论推广到带 Poission 跳的情形.另外,本文还讨论了以下形式的边值问题:dx(t) = f(t,x(t),y(t))dt y(t)dMt,Ax(0) Bx(1) =ξ*,t∈[0,1],并证明了在Lipschitz条件下适应解的存在唯一性.     

反射型的带跳倒向双重随机微分方程

证明了反射型的带跳倒向双重随机微分方程的解的存在唯一性.主要方法是Snell包和不动点定理.     

带Markov跳时变随机种群收获系统数值解的收敛性

讨论了一类带Markov跳时变随机种群收获系统的数值解问题.利用EulerMaruyama方法给出了时变种群系统的数值解表达式,在局部Lipschitz条件下,证明了方程的数值解在均方意义下收敛于其解析解.最后,通过数值例子对所给出的结论进行了验证.     

带跳的时滞随机微分方程近似解的收敛性

本文研究了一类具有Possion跳的时滞随机微分方程（SDDEwJPs）．在一般情况下SDDEwJPs没有解析解．因此合适的数值逼近法，例如欧拉法，就是在研究它们性质时所采用的重要工具．本文在局部李普希兹条件下证明了欧拉近似解强收敛于SDDEwJPs的精确解（分析解）．     

带Poisson跳和Markovian调制的中立型随机延迟微分方程的数值解的收敛性

本文研究带Poisson跳和Markovian调制的中立型随机微分方程的数值解的收敛性质.用数值逼近方法求此微分方程的解,并证明了Euler近似解在此线性增长条件和全局Lipschitz条件更弱的条件下仍均方收敛于此方程的解析解.     

带有Poisson随机测度的比例微分方程数值解的收敛性

主要研究了带跳的随机比例微分方程dX(t)=f((X(t),X(qt))dt+g(X(t),X(qt))dW(t)+∫nh(X(t),X(qt),u)N(dt,du),0≤t≤T,X(0)=X0,给出了此方程的Euler数值解,并在局部Lipschitzs条件下,证明了数值解依均方和概率测度意义下收敛于精确解.     

带跳的随机微分方程的Euler折线逼近

熟知当随机微分方程的系数不满足Lipschitz条件,而仅满足单调性条件时,我们无法用Picard迭代法证明其解的存在性. Krylov为此对Brown运动驱动的此类方程用Euler折线逼近法证明了解的存在性.本文将Krylov的结果推广到带跳的随机微分方程,证明了Euler折线逼近的收敛性.这一结果是研究带跳的随机发展方程的基础,且对随机微分方程的数值计算有用.     

The Filtering Equations of Forward-Backward Stochastic Systems with Random Jumps and Applications to Partial Information Stochastic Optimal Control

In this article, we consider a filtering problem for forward-backward stochastic systems that are driven by Brownian motions and Poisson processes. This kind of filtering problem arises from the study of partially observable stochastic linear-quadratic control problems. Combining forward-backward stochastic differential equation theory with certain classical filtering techniques, the desired filtering equation is established. To illustrate the filtering theory, the theoretical result is applied to solve a partially observable linear-quadratic control problem, where an explicit observable optimal control is determined by the optimal filtering estimation.     

Numerical Solutions of Stochastic Differential Delay Equations with Jumps

Abstract

In this article, we investigate the strong convergence of the Euler–Maruyama method and stochastic theta method for stochastic differential delay equations with jumps. Under a global Lipschitz condition, we not only prove the strong convergence, but also obtain the rate of convergence. We show strong convergence under a local Lipschitz condition and a linear growth condition. Moreover, it is the first time that we obtain the rate of the strong convergence under a local Lipschitz condition and a linear growth condition, i.e., if the local Lipschitz constants for balls of radius R are supposed to grow not faster than log R.     

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.     

局部Lipschitz条件下的布朗运动和泊松过程混合驱动的正倒向随机微分方程

本文得到在局部Lipschiz条件下的布朗运动和泊松过程混合驱动的倒向随机微分方程的存在唯一性；同时也证明了布朗运动和泊松过程混合驱动的完全藕合的正倒向随机微分方程在局部Lipschitz条件下的解的存在唯一性。     

Extrapolation of the Stochastic Theta Numerical Method for Stochastic Differential Equations

ABSTRACT

The stochastic theta method is a family of implicit Euler methods for approximating solutions to Itô stochastic differential equations. It is proved that the weak error for the stochastic theta numerical method is of the correct form to apply Richardson extrapolation. Several computational examples illustrate the improvement in accuracy of the approximations when applying extrapolation.     

Stochastic Differential Equations with Non-Lipschitz Coefficients in Hilbert Spaces

Abstract

In this article, we discuss the successive approximations problem for the solutions of the semilinear stochastic differential equations in Hilbert spaces with cylindrical Wiener processes under some conditions which are weaker than the Lipschitz one. We establish the existence and the uniqueness of the solution and additionally, in our framework we consider a limiting problem for the mild solution. It is shown that the mild solution tends to the solution of the stochastic differential equation of Itô type in finite dimensional space.     

Numerical analysis for stochastic age-dependent population equations with Poisson jumps

In this paper, stochastic age-dependent population equations with Poisson jumps are considered. In general, most of stochastic age-dependent population equations with jumps do not have explicit solutions, thus numerical approximation schemes are invaluable tools for exploring their properties. The main purpose of this paper is to develop a numerical Euler scheme and show the convergence of the numerical approximation solution to the true solution.